










EXERCISES IN TRIGONOMETRY 


BY 

DAVID RAYMOND CURTISS 

II 

. AND 

ELTON JAMES J^OULTON 

PROFESSORS OF MATHEMATICS 
NORTHWESTERN UNIVERSITY 



D. C. HEATH AND COMPANY 


BOSTON NEW YORK CHICAGO 

ATLANTA SAN FRANCISCO DALLAS 

LONDON 


EXERCISES IN PLANE TRIGONOMETRY 


This set of exercises is intended to furnish all the daily problem 
material for courses of varying length. It is designed to supple¬ 
ment a textbook or a lecture course; only enough general ex¬ 
planations are included as will suffice to make the statements of 
problems intelligible. While especially adapted to the style of 
presentation that uses coordinates from the start, its loose-leaf 
form makes possible the selection of material for any standard 
course in Plane Trigonometry. 

Each numbered problem consists of four similar exercises lettered 
(a), (6), (c), {d). Answers are given for (a) exercises, and for 
these only. In an ordinary course, but one of these four exercises 
would be assigned. All the exercises (6), for example, would 
constitute a full set for a course in Trigonometry. This plan 
makes it possible for different sections to have different assign¬ 
ments in the same year, or in successive years. 

These Exercises are so bound that the manual can be preserved 
as a whole. It is recommended, however, that the sheets be 
detached, handed in for correction, and then kept for reference in 
a loose-leaf binder, or that all be detached and so bound from the 
start. This loose-leaf plan secures freedom in the choice and 
order of material and makes possible a variety of procedures in the 
inspection and correction of problems. 

Apart from the provision of fresh problem material, the chief 
value of such a manual as this should consist in the orderly, neat, 
and uniform arrangement of problems which it encourages and 
helps to provide. It should save time and eyesight for both 
student and instructor. The authors and printers have taken 
especial pains with arrangement and typography. After the 
statement of each problem sufficient blank space is left for the 
solution, the form for the work is often indicated, and the place 
for the answer is shown. But one side of each sheet is thus used; 
the other side, left blank, may be utilized for supplementary 
solutions or additional problems, which may be others from this 
manual, or new ones assigned by the instructor. An ordinary 
lesson would consist of the material on from two to four sheets. 

The problems are mostly of medium difficulty, with many easier 
drill exercises, and a few to test the better students. The authors 
have taken pains to avoid vagueness or ambiguity in their state¬ 
ments. 

This manual should make it unnecessary to discard an excellent 
textbook because its problem supply has been exhausted. The 
fact that these problems indicate in explicit form the standard 
course as now given in American colleges and universities, and 
as taught by.the authors, will make this collection of exercises 
especially useful for self study, for preparation for examinations, 
and for correspondence courses. 


©Cli 


lA 40453 

Copyright 1931 by D. R. Curtiss and E. J. Moulton 
3 I 


D. R. C. 

E. J. M. 


NAME 


DATE 


SECTION 




1. Draw the angles whose magnitudes are given in one of the following sets: 

(а) 270°, 135°, — 225°, — 630°, 2.5 right angles. 

(б) 180°, 225°, 1170°, - 300°, 1.5 right angles. 

(c) — 270°, 315°, 1215°, — 855°, — 3.5 right angles. 

(d) — 90°, 240°, — 315°, — 1215°, — 2.5 right angles. 


2. With a 'protractor construct the angles given in one of the following sets: 


(a) 10°, 210°, - 83°, 527°, - 933°. 
(c) 40°, 110°, 262°, - 198°, - 753°. 


(b) 70°, 160°, 318°, - 482°, - 872°. 
(d) 80°, 200°, 478°, - 413°, - 1291°. 



COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


1 















NAME 


DATE 


SECTION 


3. Estimate the measure in degrees of the angles A, B,C of one of the following triangles, then measure 
them with a protractor: 



(c) (d) 


Ans. 

(a) 

Estimates A = 30°, 

B 



Measures A = , 

B 

Ans. 

(6), (c), or (d) 

Estimates A = , 

B 



Measures A = , 

B 


25°, C = 125°. 
, C' = 

, C= . 

, C = 


4. Solve one of the following problems: 

(a) Through an angle of how many degrees does the minute hand of a watch turn in 140 minutes ? 
The second hand ? The hour hand ? 

(5) Through an angle of how many degrees does the minute hand of a watch turn in a week? 
The second hand ? The hour hand ? 

(c) An auto wheel whose circumference is 8.4 ft. rolls forward 14 ft. Through an angle of how 
many degrees does a spoke turn as viewed from the left-hand side of the car ? As viewed from the 
right-hand side ? 

(d) A bicycle wheel whose circumference is 7.5 ft. rolls forward 17.5 ft. Through an angle of 
how many degrees does a spoke turn as viewed from the left-hand side of the bicycle ? As viewed 
from the right-hand side ? 

Ans. (a) - 840°, - 50,400°, - 70°. 

Ans. (b), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


3 










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NAME 


DATE 


SECTION 


In the following Exercises the direction from 0 to P is described in each of three ways ; 

I. By giving a polar angle of OP, an angle from OE to OP in 
the Figure. 

II- By giving the bearing of OP, according to the system used 
in the U. S. navy; this bearing is the angle NOP measured from 
the north through the east in the clockwise direction. 

III. By giving the surveyor’s bearing of OP; this is the acute W 
angle which OP makes with ON or OS. Thus in the Figure the 
bearing of OP is North 57° East (briefly N 57° E), and the bearing 
of OP' is S 80° W. 

In the following tables one description is given of a direction for 
each line; the student is required to supply in one of the sets the other two descriptions of the same 
direction. 


1\ 


^ __ 

0 

P' 



s 


I 

Polar Angle 

II 

Navy Bearing 

III 

Surveyor’s Bearing 


I 

Polar Angle 

II 

Navy Bearing 

III 

Surveyor’s Bearing 

- 25° 
Ans. 230° 
Ans. 160° 

Ans. 115° 
220° 

Ans. 290° 

Ans. S 65° E 
Ans. S 40° W 
N70° W 

6. (a) 

412° 

307° 

S 50° E 

75° 

130° 

S25° W 

(&) 

- 150° 

155° 

N40° W 

195° 

55° 

S35°E 

(c) 

116° 

45° 

S45°W 

- 235° 

190° 

N20° W 

id) 

- 750° 

225° 

N55°E 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

5 























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NAME DATE 


SECTION 



Rectangular coordinates of P in the Figure are 

Y 




a: = OM, y = MP. 




P 

Polar coordinates of P are 




y 

r = OP, 6 = ZMOP. 

O 

oc 

M X 

7. In the left-hand diagram below locate points whose 
polar coordinates are given in one of the following sets: 






(a) A(6, 30°), 5(10, 135°), C(4, 300°), 5(5, 270°), 5(4, - 110°), 5(8, 950°). 

(b) A(5, 70°), 5(4, 170°), C(6, 200°), 5(8, 280°), 5(5, - 180°), 5(7, 1080°). 

(c) A(7, 50°), 5(5, 145°), C(6, - 150°), 5(8, - 400°), 5(4, - 270°), 5(6, 180°). 

(d) A{8, 420°), 5(5, 210°), 5(6, - 90°), 5(7, 460°), 5(3, - 630°), 5(6, 900°). 



8. Using the 'polar axis OA in the right-hand Figure above, locate by means of a protractor and ruler 
points whose polar coordinates are given in one of the following sets; and write doion approximate 
rectangular coordinates for each point with respect to the axes OX and 0 Y: 

(a) A(7, 38°), 5(6, 247°), 5(5, - 71°), 5(7, - 585°). 

(5) A(8, 12°), 5(6, 106°), 5(12, 233°), 5(10, - 407°). 

(c) ^(5, 71°), 5(7, 171°), 5(12, - 140°), 5(6, - 758°). 

{d) A(6, 57°), 5(8, - 222°), 5(5, 263°), 5(4, 317°). 

Ans. Approximate rectangular coordinates are: 

(a) A(5.5, 4.3), 5(- 2.3, - 5.5), 5(1.6, - 4.7), 5(- 4.9, 4.9). 

(h), (c), or (d) A( ), B( ), C( ), D( ). 




COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


7 

























































































































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NAME 


DATE 


SECTION 


9. Plot on the accompanying diagram the points whose rectangular coordinates are given in one oj 
the following sets. Find for each point hy computation the value of r, sin d, cos 6, and tan where 
r, 6 are polar coordinates of the point. 

(а) A(4, 3), 5(6, - 8), C(- 9, - 12), D{5, - 12), 5(0, - 3). 

(б) A(3, 4), 5(- 8, 6), C(- 12, - 9), 5(12, - 5), 5(0, 5). 

(c) A(5, 12), 5(- 12, 5), C(- 8, - 6), 5(9, - 12), 5(5, 0). 

(d) A(20, 21), 5(- 5, 12), C(- 12, - 5), 5(21, - 20), 5(- 5, 0). 


Ans. 


(a) A 
B 
C 
5 
5 

(5), (c), or (d) A 


B 


C 


5 


5 


r 

sin 6 

COS d 

tan e 


3 

4 

3 


5 

z 

T 

10 

_ 4 

5 

3 

5 

_ 4 

3 

15 

_ 4 

5 

_ 3 

5 

4 

z 

13 

1 2 

5 

1 2 

i.O 

1 3 

1 3 

5 

3 

- 1 

0 

— 







EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

9 









































































































































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NAME 


DATE 


SECTION 


iO. For each of the following points r = 10. Compute the rectangular coordinates of each point in 
one of the following sets where sin d and cos 6 have the values given, and locate each point on the accom¬ 
panying diagram: 

(a) (h) 




COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


11 




























































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NAME 


DATE 


SECTION 


11. For one of the following sets of points, find for each point the six trigonometric functions of 6 
when the terminal line passes through the point designated: 

(а) A{9, 12), 5(7, - 24), C(- 21, 20), D{- 8, - 15). 

(б) A(4, 3), 5(20, - 21), C(- 7, 24), D{- 6, - 8). 

(c) A(8, 15), 5(- 4, 3), C(21, - 20), 5(- 16, - 30). 

(d) A(20, 21), 5(- 30, 16), C(- 8, - 6), 5(3, - 4). 


Ans. (a) (in part) 

A: r = Vo^ + 12^ = 15, sin 6 = y. = — = 

r 15 5 

a X 3 ± a V 4: 

COS 0 = - = tan 0 = ^ 

r 5 a; 3 

CSC 0 = f , sec 0 = f, cot 0 = f . 


5; sin 0 = — f|-, cos 0 = , tan 0 = 

CSC 0 = , sec 0 = , cot 0 = 


C: sin 0 = 

, cos 0 = 

, tan 0 = 


CSC 0 = 

, sec 0 = 

, cot 0 = 

• 

5: sin 0 = 

, cos 0 = 

, tan 0 = 

f 

CSC 0 = 

, sec 0 = 

, cot 0 = 

• 


Ans. (b), (c), or (d) 




A: 


B: 


C: 

D: 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

13 














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NAME 


DATE 


SECTION 


12. For one of the following sets find in each of the four cases in what quadrants 6 may terminate: 

(а) If sin 0 = — f; if cos 0 = f; if cot 0 = — 3; if sec 0 = — 1. 

(б) If cos 0 = f; if tan 0 = — 2; if esc 0 = 2; if cot 0 = 0. 

(c) If tan 0 = — f; if sin 0 = |; if sec 0 = — 2; if esc 0 = 1. 

id) If sin 0 = — I; if cos 0 = — ; if cot 0 = — 3; if cos 0 = 1. 

Ans. (a) III or IV; I or IV; II or IV; negative a;-axis. 

Ans. if}), (c), or {d) 


13. Give the signs of the six trigonometric functions of each angle in one of the following sets: 

(a) 78°, 213°, - 310°, - 601°, 1111°. 

(b) 13°, 471°, - 500°, 1000°, - 1500°. 

(c) 25°, 301°, - 140°, - 800°, 2000°. 

(d) 107°, 265°, - 513°, 640°, 3000°. 

Ans. (a) (in part) 


Angle 

in 0 

cos 0 

tan 0 

CSC 0 

sec 0 

cot 0 

78° 

213° 

- 310° 

- 601° 
1111° 

+ 1 

1 + 

+ 

+ 

+ 

+ 

or (d) 

Angle 

sin 0 

C03 0 

tan 0 

CSC 0 

sec 0 

cot 0 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


15 





c 


NAME 


DATE 


SECTION 


14. For one of the following sets find by measurement and calculation the values of the sine, cosine, 
and tangent of each angle, using the figure below: 

(a) 70°, 80°, 90°, 100°, 260°. (6) 120°, 240°, 250°, 270°, 280°. 

(c) 40°, 60°, 80°, 110°, 280°. (d) 140°, 240°, 260°, 280°, 300°. 



6 

sin d 

cos 6 

tan d 

o 

O 

00 




90° 




100° 




260° 





Ans. (6), (c), or (d) 


0 

sin 0 

cos 0 

tan 0 






















COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


17 












































































































































































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NAME 


DATE 


SECTION 


15. For one of the following sets find, hy use of the Figure of Exercise I 4 , o>ri approximate value of 6: 


(a) If sin 0 = f, cos 0 ^ If cos 0 = — 1. 


(6) If cos 0 = y\, tan 0 = — If tan 0 = 0, cos 0 positive. 

(c) If tan 0 = -1^, sin 0 = — if. If sin 0 = — 1. 

(d) If cot 0 = — If, cos 0 = — ff. If cot 0 = 0, sin 0 negative. 

Ans. (a) 0 = 143°, 180°. 

Ans. (6), (c), or (d) 


16. For one of the following sets find from suitable figures the exact values of the functions of the angles 


given : 


(a) 315°, 210°. 
(c) 150°, 225°. 


(6) 300°, 135°. 
id) 240°, 330°. 


Ans. (a) 



sin 315° = - iy2, tan 315° = - 1, 
cos 315° = fV2, cot 315° = - 1, 


sec 315° =V2, 
CSC 315° =-V2. 


sin 210° = - h 
cos 210° = - f V3, 


tan 210° = fV^, sec 210° - - 2\/3, 
i, cot 210° = V3, CSC 210 






Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E, J. MOULTON 


19 












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NAME 


DATE 


SECTION 


17. For one of the following sets find from suitable figures the exact values of the functions of the 
angles given: 

(a) - 60°, 270°. (6) 180°, - 150°. 

(c) - 90°, - 30°. Id) 540°, - 45°. 



Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


21 














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NAME 


DATE 


SECTION 


18. Prove one of the following statements: 


(а) cos 60° sin 330° — cos 30° sin 300° = 

(б) cos 30° cos 330° + sin 45° cos 225° = 1. _ 

(c) sin 45° cos 300° — cos 60° sin 225° = iV2, 

(d) sin 30° cos 300° — sin 60° cos 210° = 1. 


Ans. (a) cos 60° sin 330° 
Ans. (6), (c), or (d) 


cos 30° sin 300° = -’ ( 
2 \ 


I) 


Vs /_ 

2 A 2 J 


1 , 3 _ 1 
4 4 2 


19. For one of the following set find the values of the other functions of the angle 9, when it is given 


that: 


(a) 

CSC 9 = 

- _ 13 

5 • 





(c) 

sin 9 = 

- _ 15 

17* 





Ans, 

• (a) 








sin 9i = 

_ 5 

T¥> 

sin 

02 

— _ 5 

T¥> 



cos 9i = 

_ ] 2 

1 3> 

cos 

02 

— 12 

1 3> 



tan 9i — 

5 

1 2> 

tan 

02 

— _ 5 

1 2> 



cot 9i = 

1 2 

5 ) 

cot 

02 

— _ 12 

5 > 



sec 9i = 

— 13 

1 2> 

sec 

02 

— 13 

1 2> 



CSC 9i = 

_ 13 

CSC 

02 

— _ 13 

"5 • 

Ans, 

■ (&), 

(c), or id) 






(6) tan0 = 
id) cos 6 = 



EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

23 









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NAME 


DATE 


SECTION 


?i0. For one of the following set find the values of the other functions of the angle 6, when it is given 
i hat: 


(a) sin 6 = ^. 


( 6 ) cos 0 = 

(c) sec e = a. 


(d) tan 0 = 

Ans. 

(a) 




sin di = 

tan 01 = -|V 3 , 

sec 01 = f Vs, 


cos di = 

cot 01 = v's, 

CSC 01 = 2 . 


sin 02 = h 

tan 02 = — 3 , 

sec 02 = — fVs, 


cos 02 = — 

cot 02 = - Vs, 

CSC 02 = 2 . 

Ans. 

( 6 ), (c), or {d) 




3 

5 * 

_ 1 2 

5 • 



21. For one of the following set find the values of the other functions of the angle 6, when it is given 
ihat: 

(a) tan 0 = — |. (6) cot 6 = — 1. 

(c) sin 0 = — (d) seed = — ff. 


Ans. 

(a) 

sin 01 
cos 01 

— 3 
-E) 

~ ~ h 

tan 01 = - f, 
cot 01 = — f. 

sec 01 = - f, 

CSC 01 = f. 

II 


SS a? =4 



sin 02 
cos 02 

— _ 3 

5 > 

— 4 

5 > 

tan 02 = - I, 
cot 02 = — i, 

sec 02 = f, 

CSC 02 = — f. 


£P *-4 V 



Ans. 

(b), 

(c), or 

(d) 








COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


25 













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NAME 


DATE 


SECTION 


22. For one of the sets (h), (c), or (d) complete the equations as shown in (a), by use of Ji.-place tables: 


(a) 

sin 18° 50' = 

.3228, 

cos 37° 0' 

= 

.7986, 


tan 55° 30' = 

1.455, 

cot 42° 20' 

= 

1.098, 


sec 7° 40' = 

1.009, 

CSC 69° 10' 

= 

1.070. 

(b) 

sin 23° 0' = 

} 

cos 32° 30' 

= 



tan 15° 10' = 

J 

cot 8° 50' 

= 

>' 


sec 63° 20' = 

J 

CSC 47° 40' 

= 


(c) 

sin 37° 10' = 

} 

cos 10° 20' 

= 

7 


tan 48° 0' = 

} 

cot 9° 50' 

= 

9 


sec 51° 30' = 

7 

CSC 25° 40' 

= 



{d) sin 47° 20' = 
tan 18° 10' = 
sec 75° 0' = 


cos 5° 50' = 
cot 37° 30' = 
CSC 21° 40' = 


23. For one of the following sets find the angle A in each of the equations : 

(а) sin A = ,6361, cos A = .9750, tan A = 7.115, cot A = .3249. 

(б) sin A = .3746, cos A = .5225, tan A = 8.144, cot A = 3.305. 

(c) sin A = .5299, cos A — .9990, tan A = .1554, cot A = .4314. 

(d) sin A = .9903, cos A = .9781, tan A = .2309, cot A = 1.653. 

Ans. (a) 39° 30', 12° 50', 82° O', 72° O'. 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


27 







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Name 


DATE 


SECTION 


24. By interpolation find the values of one of the following sets: 


(a) 

sin 41° 51' = 

.6672, 

(&) 

cos 8 ° 15' = 


tan 26° 16' = 

.4935, 


sin 37° 8 ' = 


COS 11° 37' = 

.9795. 


tan 42° 24' = 

(c) 

cot 21° 17' = 

f 

{d) 

sin 20° 52' = 


sec 15° 21' = 

i 


cos 35° 14' = 


sin 35° 12' = 

• 


tan 14° 43' = 


26. By interpolation find the values of one of the following sets: 


(a) sin 48° 32' = 

.7494, 

(6) sin 51° 35' = 

tan 57° 6' = 

1.546, 

cot 68° 18' = 

CSC 52° 21' = 

1.263. 

sec 73° 11' = 

(c) tan 80° 25' = 

J 

{d) cot 79° 13' = 

sec 63° 17' = 

1 

sec 48° 5' = 

cos 70° 12' = 

• 

cos 84° 54' = 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


29 













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NAME 


DATE 


SECTION 


26. By interpolation find the value of A to the nearest minute in one set of the following equations: 

(a) sin A = .2320, cot A = 1.300. (&) tan A = .1115, cos A = .9940. 

(c) cot A = 3.245, tan A = .8446. (d) sin A = .5207, cos A = .9105. 

Ans. (a) 13° 25', 37° 34'. 

Ans. (6), (c), or (d) 


27. By interpolation find the value of A to the nearest minute in one set of the following equations: 

(a) sin A = .8100, cos A = .5105. (h) tan A = 1.127, cot A = .6845. 

(c) cos A = .4035, tan A = .4050. (d) cot A = .1540, sin A — .9500. 

Ans. (a) 54° 6', 59° 18'. 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


31 






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NAME 


DATE 


SECTION 


28. Solve one of the following right triangles and check your solution. In each case C = 90°. The 


other given parts are: 



(a) A = 27° 16', c = 

8.100. 

(6) B = 42° 20', c = 27.00. 

(c) A = 18° 34', c = 

44.00. 

(d) B = 35° 15', c = 12.00. 

Ans. (a) a = 3.711, 

h = 7.200, B = 62° 44'. 


Formulas 


Computation 

B = 90° - A, 

sin 27° 16' = .4582 

cos 27° 16' = .8889 

a = c sin A, 

c = 8.1 

8.1 

6 = c cos A. 

4582 

8889 

Check: ^ = cot B. 

0 

36656 

71112 

a = 3.71142 

b = 7.20009 


.51547 



Check; 7.2)3.71142 

cot B (from the tables) = .5154 


3 60 


The check is not perfect, 

111 


but an error of 1 in the 

72 


fourth significant figure is 

394 


permissible. 

360 



342 

288 

54 


Ans. (6), (c), or {d) 
Formulas 


Computation 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


33 



















NAME 


DATE 


SECTION 


29. Solve one of the following right triangles and check your solution. In each case C = 90°. The 
other given parts are: 

(a) A = 42° 17', h = 18. (6) B = 30° 20', a = .27. 

(c) A = 50° 12', b = .48. id) B = 63° 34', a = 6.5. 

Ans. (a) B = 47° 43', a — 16.37 (= 16 to 2 significant figures), c = 24.34 (= 24 to 2 places). 
Ans. (6), (c), or (d) 


30. Solve one of the following right triangles and check your solution. In each case C = 90°. The 
other given parts are: 

(a) A = 62° 54', a = 9.2. (b) B = 48° 15', b = 83. 

(c) A = 55° 37', a = .67. (d) B = 34° 18', b = .41. 

Ans. (a) B = 27° 6', b = 4.708 (= 4.7 to 2 significant figures), c = 10.33 (= 10 to 2 places). 
Ans. (6), (c), or {d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


35 








NAf/IE 


DATE 


SECTION 


31. Solve one of the following right triangles and check your solution. In each case C = 90°. The 
other given parts are: 


(a) a = .66, b = .48. 

(c) a = 54, 6 = 81. 

Ans. (a) A = 53° 59', B = 36° 1', c = 
Ans. (6), (c), or (d) 

(6) a = 1.2, h = 2.4. 

(d) a = 35, 6 = 27. 

.8158 (= .82 to 2 significant figures). 


32. Solve one of the following right triangles and check your solution. In each case C = 90°. The 
other given parts are: 


(a) a = 1.02, c = 3.04. 

(c) a = 46.2, c = 57.3. 

Ans. (a) A = 19° 36', B = 70° 24', h 
Ans. (6), (c), or (d) 

(6) h = 506, c = 817. 

(d) h = 25.1, c = 41.6. 

= 2.864 (= 2.86 to 3 significant figures). 


EXERCISES IN PLANE TRIGONOMETRY 

COPYRIGHT 1931 BY D. R. CURTISS AND E. J, MOULTON 

37 






I 








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' 1,.'’. ■* '““T^hn. ' -'• 





NAME 


DATE 


SECTION 


33. Solve one of the following right triangles and check your solution. In each case C — 90°. The 
other given parts are: 

(a) a = 40.55, h = 35.62. (6) a = 26.61, h = 15.07. 

(c) a = 5.725, h = 4.125. (d) a = 4.027, h = 6.153. 

Ans. (a) A = 48° 42', B = 41° 18', c = 53.97, by use of 4-place tables. 

Ans. (6), (c), or (d) 


34. The data below are parts of isosceles triangles ABC, A and B being the equal base angles and C 
the vertex angle. Solve one triangle and check your solution. 

(a) A = 46° 22', c = 1262, find C, a, and b. 

(b) C = 62° 18', c = 1072, find A, a, and b. 

(c) a = 1504, c = 1240, find A, c, and b. 

(d) A = 51° 15', a = 10.51, find C, c, and b. 

Ans. (a) C = 87° 16', a = b = 914.3 by use of 4-place tables. 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


39 





m2 




wcjtm 


y\m. iDih 6^0 ^SifjS^ ^ ** jj 


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4 ^d.91* ^ n (4) 
fiSU 5^ 4^t5A}.t ,^ 4 (V>) 












NAME 


DATE 


SECTION 


35. The following problems refer to regular pentagons. In one of them find the parts asked for by 
means of those given: 

(a) The side is 4.56 in., find the radii of the inscribed and circumscribed circles. 

(5) The apothem is 8.14 in., find the perimeter and the radius of the circumscribed circle. 

(c) The radius of the circumscribed circle is 6.62 in., find the length of a side and the radius of the 
inscribed circle. 

(d) The perimeter is 25.50 in., find the apothem and the radius of the circumscribed circle. 

Ans. (a) 3.137 in. (= 3.14 in. to 3 significant figures), 3.878 in. (= 3.88 in. to 3 significant figures). 
Ans. (5), (c), or (d) 


36. Solve one of the following problems: 

(a) The angle of elevation of the top of a flagpole measured at a point on the same level as the 
base of the flagpole and 153 ft. from it is 22° 30'. How tall is the flagpole? 

(5) A vertical marker on a sundial is exactly 3 in. high. What is the elevation of the sun when 
the shadow cast is 5.3 in. long? 

(c) In a lighthouse, at 58.2 ft. above water level, the angle of depression of a small boat is 10° 30'. 
How far is the observer from the boat? 

(d) Two trees stand at points A and B, separated by a deep ravine. In order to find the distance 
between them a line AC, at right angles to AB, is measured. If AC is 102 ft. long and the angle 
ACB is found to be 42° 20', how far apart are the trees? 

Ans. (a) 63.37 ft. (= 63.4 ft. to 3 significant figures). 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D, R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


41 






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NAME 


DATE 


SECTION 


37. Solve one of the following problems: 

(a) A surveyor wishes to measure the distance between two points A and B at opposite ends of a 
lake. To do this he runs a line AC, 171 yd. long, and a line CB at right angles to AC, 205 yd. 
long. What is the distance AB? 

(h) A ladder 15 ft. long is leaned against a wall. How high on the wall will it reach when the foot 
is 8.5 ft. from the base of the wall, if the ground is level? What angle will the ladder make with 
the ground? 

(c) Two towns are connected by a right-angled system of roads. To go from one town to the 
other one must go 8 miles east and 23 miles south. How far apart are the towns ? 

(d) A guy wire of a telephone pole is 25 ft. long; one end is attached to the pole 17 ft. above the 
ground. Find the distance of the other end of the wire from the base of the pole, if this end is 
attached to a post at the level of the ground, the ground being level and the pole vertical. 

Ans. (a) 266.9 yd. (= 267 yd. to 3 significant figures). 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


43 









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NAME 


DATE 


SECTION 


38. Solve one of the following problems: 

(а) The top of a ladder is placed at a window 21 ft. above the ground. What angle does the 
ladder make with the ground if the foot is 11 ft. from the house (we assume the ground to be level) ? 
How long is the ladder if 1 ft. 6 in. projects above the point of contact with the window sill, in this 
position? 

(б) An observer at the same level as the bottom of a flagpole finds the angle of elevation of the top 
to be 27° 20'. If the flagpole is known to be 65.2 ft. tall how far is the observer from the base of the 
pole ? From the top ? 

(c) A shaft is designed to descend 35 ft. for each 100 ft. measured along the shaft. What angle 
will it make with the horizontal? When the shaft is 152 ft. long how far is it below the starting 
level ? How much horizontal distance has been traversed ? 

(d) A right triangle with sides 1.32 and 2.41 inches respectively is inscribed in a circle. What 
is the diameter of the circle ? What are the acute angles of the triangle ? 

Ans. (a) 62° 21', 25.2 ft. 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


45 










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(a) miA 


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DATE 


SECTION 


39. Solve one of the following problems: 

(a) A and B stand on a level plain and sight at the same instant at a balloon directly above a 
point on the line joining them. The angle of elevation of the balloon is 35° 20' at A’s station and 
53° 10' at B’s station. If the two stations are exactly a mile apart how high was the balloon at the 
time of observation? 

(b) From one point of observation, on a level with the base of a hill, the angle of elevation of the 
top is 44°. From a point 252 yd. farther away on the same level surface the elevation of the top 
is 35°. How high is the hill ? 

(c) A flagpole is mounted on the top of a wall. At a point level with the base of the wall and 
25.5 ft. away, the angle of elevation of the bottom of the pole is 37° 10', and of the top is 54° 20'. 
What is the height of the pole ? 

(d) A power plant has built a new smokestack. At a point 252 ft. from the new stack and 176 ft. 
from the old, the angle of elevation of the top of each is 38° 30'. If the point of observation is in 
the same horizontal plane as the base of each stack, how much taller is the new stack than 
the old? 

Ans. (a) 2435 ft. 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


47 









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i V *65 ii'4«tr U h iniT 

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( 


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(#1 jpA 

iv- _ J«A 







NAME 


DATE 


SECTION 


40. By use of a suitable figure reduce each expression in one of the following sets to a function of an 
acute angle: 


(a) sin 170°, cos 250°, tan (— 75°). 
(c) tan 315°, sin 225°, esc (- 120°). 



(6) cot 420°, sec (- 125°), cos 255°. 
(d) cos (— 150°), tan 350°, sin 215°. 




In each case the angles M'OP' and MOP are equal. The lengths of OP' and OP are taken 
equal, making the right triangles OM'P' and OMP equal in each figure. 

(1) y' = y,r' = r, (2) x' = - x, r' = r, 

sin 170° = ^ = ^ = sin 10°, cos 250° = % = — - = — cos 70°, 

r r r r 

(3) y' =- y,x' = X, 

tan (- 75°) = ^ = - tan 75°. 

X X 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


49 

































NAME 


DATE 


SECTION 


41. By proper application of the rule: 

Given function of (n • 180° ± d) — ± same function of 6, 

reduce each expression in one of the following sets to a function of an acute angle: 

(a) sin 1000°, sec (- 310°), tan 570°, cos (- 280°), cot 175°. 

(5) cos (— 370°), CSC 515°, cot (— 150°), tan 410°, sin 560°. 

(c) tan 225°, sin 470°, sec (— 85°), esc 835°, cos (— 170°). 

Id) cot 390°, cos (- 400°), esc 745°, sin (- 275°), tan 165°. 

Ans. (a) sin 1000° = sin (6 • 180° — 80°) = — sin 80°, 

sec (- 310°) = sec (- 2 • 180° + 50°) = sec 50°, 

tan 570° = tan (3 • 180° + 30°) = tan 30°, 

cos (- 280°) = cos (- 2 • 180° + 80°) = cos 80°, 

cot 175° = cot (180° - 5°) = - cot 5°. 

Ans. (b), (c), or {d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


51 







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NAME 


DATE 


SECTION 


42. By means of the tables find the value of each expression in one of the following sets: 

(а) sin 125° 10', cos 215° 20', esc 530°, tan (- 80°), sec 1027°, cot 755° 12'. 

(б) cos 565° 18', tan (- 227°), sin 1152° 10', cot 310° 15', esc 862° 30', sec 477°. 

(c) tan (- 1873°), sin (- 420°), cos 710° 22', sec 515° 18', esc 627° 20', cot 456°. 
id) cot 371° 25', tan 821° 40', sec (- 410°), sin 356° 6', esc 700°, cos 465° 56'. 

Ans. (a) .8175, - .8158, 5.759, - 5.671, 1.662, 1.417. 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


53 








(i 


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rt^''''ifor,r»ovn MuKti »i s^nxtxM 


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43. For one of the following cases find an angle 6 terminating in the given quadrant and satisfying 
the given equation: 

(a) 2nd quadrant, sin 6 = .8274. (6) 3rd quadrant, tan 6 = 1.144. 

(c) 3rd quadrant, cos 0 = — .7844. (d) 4th quadrant, cot 0 = — 2.747. 

Ans. (a) 123° 10'. 

Ans. (6), (c), or (d) 


44. For one of the following cases find an angle d terminating in the given quadrant and satisfying 
the given equation: 

(a) 3rd quadrant, cos 6 = — .9290. (6) 2nd quadrant, cot 6 = ~ .5900. 

(c) 4th quadrant, sin 0 = — .2580. (d) 3rd quadrant, tan^ = 1.240. 

Ans. (a) 201° 43'. 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

55 

















NAME 


DATE 


SECTION 


45. For one of the follovnng sets find the rectangular coordinates of the points whose polar coordinates 
are given: 

(a) (2, 145®), (.5, - 110®), (3, 315® 10'), (10, 157° 12'). 

(b) (1, 343® 26'), (.2, - 221® 10'), (4, 163®), (5, 247°). 

(c) (6, - 111®), (4, 157® 6'), (.5, 257® 10'), (2, 310® 20'). 

(d) (2, - 97°), (7, 250® 14'), (8, 132°), (.2, 277® 40'). 

Ans. (a) (- 1.6384, 1.1472), (- .1710, - .4698), (2.1276, - 2.1150), (- 9.218, 3.876). 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


57 




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rw^ .(’srf ^ti ,ot) ,coi w. ,b) ,Cari ~ .a.) .s) (^0 ^ 

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NAME 


DATE 


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46. For one of the following sets find the polar coordinates of the points whose rectangular coordinates 
are given: 

(a) (2, - 2), (- 5, 5), (- 2, - 4), (- 1, 3). (h) (1, - 5), (- 3, 3), (7, - 8), (- 5, - 7). 

(c) (2, - 5), (- 2, 3), (- 8, 12), (- 4, - 4). (d) (- 1, - 1), (12, - 6), (- 3, 5), (3, - 7). 

Ans. (a) (2\/2, 315°), (5\/2, 135°), (4.472, 243° 26'), (3.162, 108° 26'). 

Ans. (6), (c), or (d) 



COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


59 





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47. By reference to a suitable figure prove one of the following sets of relations: 

(а) cos 325° = sin 55°, tan 265° = cot 5°, sin (— 830°) = — cos 20°, cot (— 210°) = — tan 60°. 

(б) sin (- 155°) = - cos 65°, cos (- 397°) = sin 53°, tan 456° = - cot 6°, sec 627° = - esc 3°. 

(c) CSC (— 252°) = sec 18°, tan 1000° = — cot 10°, cos (— 600°) = — sin 30°, sin 854° = cos 44°, 

(d) tan 1105° = cot 65°, cos (— 500°) = — sin 50°, sin 679° = — cos 49°, esc (— 313°) = sec 43°. 

Ans. (a) 



In each figure OP is taken equal to OP', the angles MOP and M'P'O are equal. It follows that 
the right triangles AIOP and M'OP' are equal. Hence 

(1) x' = y, r' = r, and cos 325° = ^ = ^ = sin 55°. 

r r 

(2) y' = — X, x' = — y, and tan 265° = — —- = cot 5°. 

(3) y' = — X, r' = r, and sin (— 830°) = ^ = —- — — cos 20°. 

(4) x' = — y, y' = X, and cot (— 210°) = —, = —- = — tan 60°. 

y X 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


61 























AiiK- « ,^«6lS;- ) rOiTaVi »- ^« CO^ -'f ah ,V m =* tiet ,<ti6 <>«. « /x>^ (is) 

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’\ 



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fMl 8waltd\ 11 .k»Jp9 aiu 0 ‘V‘U bii»< HO\rv a/vtiiitr ud> 4.'"U> ul k»n>'' twlBi ei *10 wujjd <(a*o nl 
p iiMi-n .kfip-* mbojt *\0V, «al8ftj»nJ 

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NAME 


DATE 


SECTION 


48. By reference to the rule: 

Given function of {n • 90° ± 0) = + cofunction of 6 {n odd), 

prove one of the following sets of relations: 

(а) sin (— 420°) = — cos 30°, tan (— 260°) = — cot 10°, cos 477° = — sin 27°, sec 614° = — esc 16°. 

(б) cos 970°= - sin 20°, esc (- 175°) = - sec 85°, cot 583° = tan 47°, sin (- 874°) = - cos 64°. 

(c) tan (— 435°) = — cot 15°, sec 1234° = — esc 64°, cos (— 920°) = — sin 70°, sin 663° = — cos 33°. 

(d) cot 481° = — tan 31°, cos 387° = sin 63°, sin (— 372°) = — cos 78°, tan (— 231°) = — cot 39°. 

Ans. (a) sin (— 420°) = sin (— 5 • 90° + 30°) = — cos 30°, 

tan (- 260°) = tan (- 3 • 90° + 10°) = - cot 10°, 

cos 477° = cos (5 • 90° + 27°) = - sin 27°, 

sec 614° = sec (7 • 90° — 16°) = — esc 16°. 

Ans. (6), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


feXERCISES IN PLANE TRIGONOMETRY 


63 
















NAME 


DATE 


SECTION 


49. By means of the tables fend the tatue ef each expre^^wm m «me ef ike foUatemg 

(a) sin 137®, cos 950®, tan (— 267®), sec 392®, esc 42S® lO', co« 616® 16'. 

(h) cos 287® 15', sin (— 377® 200, cot 620®, esc (— 266^. sec 731®, tan 1111*. 

(c) tan 456®, cot 365° 26', sin (— 472°^, sec 923® 20', esc 527®, cos (— 301*). 

(«0 cot (— 487®), sec 3S9® 4S', esc 925®, 65S®, cos 519°, tan 176® 20’. 

Ans. (a) .6820, - .6128, - 19.08, 1.179, 1.077, ‘2.444. 

Ans. (6), (c), or (d) 


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NAME 


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60. By reference to the rule: 

Given function of (n • 90° + 0) = + cofunction of 6 (n odd), 

prove one of the following sets of relations: 

(а) cos (— 90° — 6) = — sin 6, tan {6 — 450°) = — cot 6. 

(б) sin (— 450° — 6) = — cos 6, cot {d — 90°) = — tan d. 

(c) tan (— 90° — 6) = cot 6, sin {d — 270°) = cos d. 

(d) tan (— 270° — 0) = cot d, sec {6 — 90°) = esc 6. 

Ans. (a) cos (—90° —0) = cos ( —1 • 90°—0) = — sin d, tan (0—450°) = tan ( —5 • 9O°+0) = — cot 6. 
Ans. (5), (c), or (d) 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


67 








Fiiv 




' (% ifc ^' »5 V> noiioiutl m^lO % 

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NAME 


DATE 


SECTION 


61. Describe the variation of one of the following trigonometric functions: 

(a) cos 6. (h) sin d. (c) tan 6. (d) sec 6. 


Ans. (a) 

Ans. (6), (c), or (d) 


As 6 varies 
from . . 

0° to 90° 

90° to 180° 

180° to 270° 

270° to 360° 

cos 6 varies 
from . . 

1 to 0, 
decreasing 

0 to — 1, 
decreasing 

- 1 to 0, 
increasing 

0 to 1, 
increasing 







EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

69 



































NAME 


DATE 


SECTION 


62. By using line values, with suitable figures, prove one of the following identities 

(a) cos (180° — 6) = — cos d. (6) sec (180° + 0) = — sec 6. 

(c) tan (270° — d) = cot d. {d) sin (360° — 6) = — sin 6. 



EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

71 













^ - * (% i- 'ym) ceIb (&) i<»»»(n“ot^ aia, t») 

■ ' 

(of>^flA 

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NAME 


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SECTION 


53. Draw the graph of one of the following functions and discuss the variation of that function hy means 
of the figure: 

(a) tan x. (6) cos x. (c) sec x. (d) esc x. 

Ans. (a) The graph is shown below. When x increases from 0° to 90°, tan x increases con¬ 
tinuously from zero with no upper limit. As x increases from 90° to 180°, tan x increases from an 
indefinitely large negative amount to zero, etc. Tan x has a period of 180°. 



Ans. (6), (c), or (d) 



COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 


EXERCISES IN PLANE TRIGONOMETRY 


73 





































































































































































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NAME 


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54. Draw a graph of one of the following expressions. Has it a period? 

(a) 3 cos 2 X. (b) 2 cos 3 x. (c) 2 sin 2 x. (d) 2 sin 3 x. 

Ans. (a) 3 cos 2 x has a period of 180°. Its graph is given below. 


Y 



Ans. 


(6), (c), or (d) 



EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

75 































































































































































































































































































































NAME 


DATE 


SECTION 


55. For one of the cases helow find by using trigonometric formulas the other five functions of an 
angle 6 terminating in the quadrant specified and having one function as given : 

(a) sin 6 = 1st quadrant. (&) cos ^ 4th quadrant. 

(c) seed =- 2nd quadrant. (d) tan 6 = ^, 3rd quadrant. 

Ans. (a) cos 0 = I-, tan 0 = Vs, cot d = iVs, sec 6 = 2, esc 6 = fVs. 

Ans. (6), (c), or (d) 


56. Reduce one of the following expressions to another containing hut one.function as indicated. 
Simplify the results by algebraic means where this is possible: 

(a) Express cos^ 0 + sin 0 in terms of sin 6 only. 

(&) Express sin 6 sec 6 in terms of cos 6 only. 

(c) Express tan 6 sec d in terms of sin d only. 

(d) Express — 92^ — -f- tan d in terms of cos 0 only. 

1 + sm 0 

Ans. (a) 1 + sin 0 — sin^ 6. 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J, MOULTON 


77 








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, < .* 










NAME 


DATE 


SECTION 


57. Prove one of the following identities: 
(a) sin 6 — tan 6 = tan 6 (cos d — 1). 


(c) 


sin^ d — cos^ 6 


= sin 0 + cos 6. 


sin 0 — cos 

Ans. (a) Three methods of procedure are illustrated. 


(6) cot 0 cos 0 + sin 0 = esc 0. 

(d) ^ ^ = sin X + cos X. 

sec X 


(1) Change the left member only: 

sin 0 sin 0 cos 0 — sin 0 


sin 0 — tan 0 = sin 0 — 


cos 0 


cos 0 


sin 0 
cos 0 


(cos 0 — 1) = tan 0 (cos 0 — 1). 


(2) Change both members: 


sin 0 — tan 0 

■ n sin 0 

sin 0 —- 

cos 0 


tan 0 (cos 0 — 1) 
sin 0 , 


cos 0 


(cos 0 — 1) 

sin 0 
cos 0 


(3) Use methods of simplifying equations: 

sin 0 — tan 0 = tan 0 (cos 0 — 1) 

.. . „ sin 0 sin 0 (cos 0 — 1) 

if sin 0 --- =-^-r- 

cos 0 cos 0 

which is true if 

sin 0 cos 0 — sin 0 = sin 0 (cos 0 — 1), 
which is an identity. 


Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

79 















A tm ^ ^nki ^ K» t^) ^ , .fl - « m) ^ o»t •* < ha! - (») 


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NAME 


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SECTION 


68. Prove one of the following identities: 

(a) tan d — cot 9 = (sec 9 + esc 0)(sin 9 — cos 9). Cb) ^ ~ ^ = cos 9 — sin 9 ^ 

1 + tan 9 cos 9 sin 9 

(c) sec2 9 + csc^ 9 = sec^ 9 csc^ 9. (d) tan^ 9 - sin^ 9 = tan^ 9 sin^ 9. 


69. Prove one of the following identities: 


(a) cot A — CSC A = 
(c) sec^ A + tan^ A ■■ 


cos A — 1 ^ 

sin A 

1 + sin^ A 
1 — sin^ A 


(Jo) tan A — sec A = sec A (sin A — 1). 
{d) cos^ A csc^ A = csc^ A — 1. 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 
81 











I 



4 




















^.rvME 


DATE 


SECTION 


60. Prove one of the following identities: 

(a) sin^ X (sec^ x + csc^ x) = sec^ x. 

. s 1 + tan2 a: _ , „ 

1 + cot^ a; " 


(&) 

id) 


sec^ X 


1 + cot^ X 
sec 6 + tan 6 


= sec^ a; — 1. 

1 + sin 0 


sec 6 — tan 6 1 — sin 6 


61. Prove one of the following identities: 


(a) Y- 

(c) ( 


tan X 
— cot X 
1 + tan X 
sec X 


1 — COS^ X 
cos X (sin X — cos x) 

= 2 sin a; cos a; + 1. 


(b) 


sec a; + CSC a; 
1 + tan X 


CSC X. 


(d) tan^ X + sec^ a; + 1 = —-— 

cos^ X 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R, CURTISS AND E. 

83 


■. MOULTON 

















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b%» % 











NAME 


DATE 


SECTION 


62. Prove one of the following identities, 
sin d 


(a) cot 6 + 


1 + cos d 


= CSC 6. 


(&) 


sin^ X 


— 1 = — cos X. 


1 + cos X 

(c) 2 (1 + sin a:)(l — cos a:) = (1 4- sin x — cos xy. 

(d) sin a: (1 + sec x) — cos a: (1 + tan x) = tan x — cos x. 




63. Prove one of the following identities: 

sin X I cos X f , N/i . N 

W —~ + 7 - = (sec X + CSC a:)(l — sin x cos x), 

cot X tan X ' 

(b) tan9 + sin« = 

tan 0 — sin 0 

(c) sec^ X — tan^ x = sin^ x + cos^ x. 

(d) cos^ X = sin^ x (csc^ a; — 1). 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

85 













ill .£3 


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NAME 


DATE 


SECTION 


64. Prove one of the following identities: 
(a) tan x (cot x — cos x sin x) = cos^ x. 
(c) (tan^ X + l)(sin2 x — 1) = — 1. 


(h) cot X _ tan X 
cot^ X — 1 1 — tan^ X 

(d) (tan X + CSC xY = tan^ x + csc^ x + 2 sec x 


66. Prove one of the following identities: 

(a) (cot A —cos A)(l+sin A) = cot A cos^ A. 
sm^ cc 

(c) -+ sin X cos X = tan x. 

cos X 


(b) (sin A — CSC A)2 = cos^ A cot^ A. 

/ jx sin X + cot X . 1 + cos x — cos^ x 

{d) -!- = cot X • —- 

cos X + tan x 1 + sin x — sin^ x 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J, MOULTON 

87 



















NAME 


DATE 


SECTION 


66. By use of the addition formulas hut without using tables find the value of one of the following 
expressions: 

(a) sin 165°. (6) sin 105°. (c) cos 105°. (d) cos (- 15°). 

Ans. (a) sin 165° = sin (120° + 45°) = sin 120° cos 45° + cos 120° sin 45° 

= iVs • iV2 - i • ^V2 = i(\/6 - V2). 

Ans. (6), (c), or (d) 


67. Apply the addition formulas to one of the following expressions and reduce to numerical values, 
checking results: 

(a) cos (270° - 45°). (h) sin (90° - 45°). (c) cos (60° + 30°). (d) sin (90° + 30°). 

Ans. (a) cos (270° - 45°) = cos 270° cos 45° + sin 270° sin 45° = 0 + (- 1) • ^^2 = - iVi 
cos 225° = - iV2. 

Ans. (b), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

89 





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68 . In each of the following cases a and /3 are positive acute angles: 

(a) sina = f, cos /3 = ^ 3 , find sin (a + /3), cos (a - /3). 

(&) sin O' = f, cos ^ find sin (a - / 3 ), cos (a + ff). 

(c) sin a = f, cos jS = f|, find sin (a — ^), cos (a + | 3 ). 

(d) sin O' = cos iS = f|, find sin (a + / 3 ), cos (a - / 3 ). 

AUb. \a) gg, g^, 

Ans. (6), (c), or (d) 


69. 

In each of the following 

cases a and /3 

are 

positive acute angles 

{a) 

sin a = 

- 8 
17 > 

cos /3 = 

3 

57 

find 

sin 

(« 

— 

/3), 

cos 

(« 

+ 


(h) 

COSo; = 

-15 
17 7 

sin ^ = 

4 

57 

find 

sin 

(a 

+ 


cos 

(« 

— 


(c) 

sin a = 

- 7 

2 57 

cos /3 = 

5 

1 3> 

find 

sin 

(a 

+ 


cos 

(« 

— 


id) 

cos a = 

-24 

2 57 

sin j8 = 

1 2 

1 3) 

find 

sin 

(a 

— 

/3), 

cos 

{a 

+ 


Ans 

;. (a) 

_ 3 6 

8 £ 

; 1 3 

0 8 5* 











Ans. (6), 

(c), or (d) 












EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J, MOULTON 

91 









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NAME 


DATE 


SECTION 


70. By use of tables find the approximate difference between the expressions in one of the following 
cases : 


ia) sin (37° — 22°) and sin 37° — sin 22°. 

(&) cos (37° + 22°) and cos 37° + cos 22°. 

(c) sin (56° — 40°) and sin 56° — sin 40°. 

(d) cos (38° + 34°) and cos 38° + cos 34°. 

Ans. (a) sin (37° — 22°) = sin 15° = .2588 
sin 37° = .6018 
sin 22° = .3746 
sin 37° - sin 22° = .2272 
Ans. (6), (c), or (d) 


sin (37° - 22°) = .2588 
sin 37° - sin 22° = ,2272 
Required difference = .0316 


71. Prove one of the following identities: 

cos 0 + sin 0 


(a) sin (45° + 0) = 
(c) sin (30° -6) = 


\/2 

cos 6 — Vs sin d 


(b) cos (60^^ 
(d) cos (45^^ 


e) 


-d) = 


cos 6 + V3 sin d 
2 

cos 9 + sin 6 ^ 

V2 


Ans. (a) sin (45° + 6) = sin 45° cos 6 + cos 45° sin 0 = ^ cos 0 + ^ sin 6 = — 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

93 












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72. Prove one of the following identities: 

(а) cos (A + B) cos B + sin (A + B) sin B = cos A. 

(б) sin (A — B) cos B + cos (A — B) sin B — sin A. 

(c) cos (A — B) cos A + sin (A — B) sin A = cos B. 

(d) sin (A + B) cos A — cos (A + B) sin A = sin B. 

Ans. (a) We give two solutions. 

(1) Let a = A + B. 

Then we are to prove cos a cos 5 + sin a sin B = cos A. 

The left member is cos (a — B). By definition of a, we have a — B = A. The left member is 
therefore cos A, and the identity is proved. 

(2) Working with the left member, we have 
cos (A + B) cos B + sin (A + B) sin B 

= (cos A cos jB — sin A sin B) cos B + (sin A cos B + cos A sin B) sin B 
= cos A cos^ B — sin A sin B cos B + sin A sin B cos B + cos A sin^ B 
— cos A (cos^ B + sin^ B) 

= cos A, which is the right member. 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

95 




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NAME 


DATE 


SECTION 


73. Prove the formulas for sin (a + jS) and cos (a + jS) using two figures as specified below: 

(а) When a and /3 are each between 90° and 180°, a + jS less than 270°; and when a is between 90° 
and 180°, is acute, a + greater than 180°. 

(б) When a is between 135° and 180°, /3 less than 45°, a + jS less than 180°; and when a + /S is 
greater than 180°. 

(c) When a and jS are each between 90° and 180°, a + /S greater than 270°; and when a is between 
180° and 270°, /3 is acute, a + /S greater than 270°. 

id) When a is between 270° and 360°, /3 is acute, a + less than 360°; and when a + /3 is greater 
than 360°. 


Ans. (a) The figures are given below. The student must supply necessary equations for the 




Hint. Make necessary alterations, if any, in the proofs given for the case when a and ^ are both acute. 
Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

97 















aiLj.9 



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NAME 


DATE 


SECTION 


74 . Prove the formulas for sin (a — jS) and cos (a — /3) using two figures as specified below: 

(a) When a is between 90° and 180°, /3 acute, a — /3 less than 90°; and when a — /3 is greater 
than 90°. 

(b) When a is between 270° and 360°, /3 acute, a — /3 greater than 270°; and when a — /3 is less 
than 270°. 

(c) When a is between 180° and 270°, /3 acute, a — ^ less than 180°; and when a — /3 is greater 
than 180°. 

(d) When a is between 270° and 360°, jS between 90° and 180°, a — /3 greater than 180°; and when 
a — j8 is less than 180°. 

Ans. (a) The figures are given below. The student must supply the necessary equations. 



Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

99 














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NAME 


DATE 


SECTION 


76. By use of addition formulas hut without use of tables find the value of one of the following expres¬ 
sions, equating the angles to a sum or difference of 45° and 30°: 


(a) tan 75°, (6) tan 15°. (c) tan — 15°. 

Ans. (o) tan 75° = tan (45° + 30°) = 45° + tan 30° 

^ ^ ^ ^ 1 - tan 45° tan 30° 

^ (3 +V3)^ ^ 9 + 6V3 + 3 ^ 2 +\/3 
9-3 6 

Ans. (6), (c), or (d) 


(d) tan (— 75°). 
1 +iV3 _ 3 + V3 
1 - iVd 3 - V3 


76. Find the values of tan {x + y) and tan {x 
In each case x and y are positive acute angles: 

(a) sin X = tan y = 

(c) tan X = cosy = |. 

Ans. (a) If, If. 

Ans. (6), (c), or {d) 


y) under one of the following sets of conditions. 

(6) cos X = If, tan y = ^. 

(d) tan X = V = 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 
101 












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NAME 


DATE 


SECTION 


77. Prove one of the following identities: 

(a) tan (0 - 45°) = ~ (6) tan (0 + 60°) 

1 + tan 0 V / V / 

(c) tan (30° - 0) = 1 -V'3tan0 ^ 

V3 + tan 0 

Ans. (a) tan (0 - 45°) = Jan 0 - tan 45° ^ tan 0 - 1 ^ 

1 + tan 0 tan 45° 1 + tan 0 

Ans. (6), (c), or {d) 


tan 0 + Vs 
1 — Vs tan 0 
Vs — tan 0 
1 + Vs tan 0 


78. 

(a) 

(c) 


Prove one of the following identities . 
tan {x -p y) — tan x 


1 + tan {x + y) tan x 
cot (a — j3) cot <3 — 1 
cot (a — |8) + cot /3 

Ans. (a) 

Ans. (6), (c), or {d) 


= tan y. 


= cot 


(&) 


tan {y — x) tan x 


1 — tan {y — x) tan x 
_ cot {a + /3) cot j3 + 1 
cot (« + /3) — cot i8 


= tan y. 


= cot a. 


tan(x + y) - taax ^ 
1 + tan {x + y) tan x 


x] = tan y. 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

103 


























NAME 


DATE 


SECTION 


79. Find the values of sin 2 a, cos 2 a, tan 2 a, sin cos tan without using the tables, from one 

2 2 2 

of the following data, a being acute: 

(a) sin a = |. (5) sin a = i%. (c) sin a = H• («) sin a = 

Ans (n^ ^ — _7 —24 iv/^ 2 \/k 1 

2~S) "T~> "S'* 

Ans. (6), (c), or ((i) 


80 . Using the half angle formulas make one of the following substitutions: 

(a) ot 45 j find v9;lu.GS of sin0^ cosiriGj s^nd of 22^^ . 

(b) a — 150°; find values of sine, cosine, and tangent of 75°. 

(c) a — 135°; find values of sine, cosine, and tangent of 67^°. 

id) a. — 225°; find values of sine, cosine, and tangent of 112^°. 

Ans. (a) sin 22^° = 1 V 2 -V2; cos 22^° = ^V2 +V2; tan 22^° = V2 - 1. 
Ans. (6), (c), or (d) 


EXERCCSES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

105 












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tsx^onooirr «uun 






NAME 


DATE 


SECTION 


81. Prove one of the following identities. 
(a) (sin d - cos 0)2 = l _ gin 2 6. 
f-. 2 cos X 

\C) - -= CSC X. 


/, X sin 2 X 
(o) - = cos X. 


id) 


2 sin X 
cos 2 X 


sin X + cos X 


sin 2 X 

Ans. (a) Working with the left member, we have 

(sin 6 — cos 0)2 = sin2 0 — 2 sin 0 cos 0 + cos^ 0 
= (sin2 0 + cos2 0) — 2 sin 0 cos 0 
= 1 — sin 2 0. 


= cos X — sin X. 


82. Prove one of the following identities: 


(a) 


tan 2 a 
2 tan a 


cos2 a 
cos 2 a 


(c) tan 2 A (1 + tan A) 


2 tan A 
1 — tan A 


(h) 

id) 


cot a — tan a 
2 


cot 2 a. 


cos 2 A 
2 cos A 


cos A — 


sec A 
~2 


I 

I 

\ 

i 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D, R. CURTISS AND E. J. MOULTON 

107 















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NAME 


DATE 


SECTION 


83. Prove one of the following identities: 


(a) 

(c) 


tan X = 


1 — cos 2 X 


sin 2 X 

cos {x y) = sin x cos y (cot x — tan y). 


(6) sin* X cot* X — cos* x tan* x = cos 2 x. 
(d) sin a sin 2 a = cos a (1 — cos 2 a). 


84. Prove one of the following identities: 


(a) 

(c) 


sin* X — sin* y . 
sin {x — y) 

cot X — cot 2 X 


sin {x -\-y). 

sec X CSC x^ 
'' 2 ’ 


(5) tan - — tan x = esc x (1 — sec x). 


{d) cot X = 


1 + cos 2 X _ 

2 sin X cos x 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

109 








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NAME 


DATE 


SECTION 


85 . 

(a) 

(c) 


Prove one of the following identities: 
cot 2 aj = i cot X — ^ tan x. 


sec X + tan y esc x 
CSC X — tan y sec x 


tan {x -\- y). 


( 6 ) 1 
id) 1 


2 ^ 

— COS^ - = 

2 

— tan X 


1 — cos d 


sin 2 a: + cos 2 a;- 
2 sin X cos x 


86. Prove one of the following identities: 

(a) ^ — = (sec 2 a: + tan 2 a;)^. 

' 1 — sin 2 aj 

(c) tan ^ sin a; = 2 sin^ 


(&) 

id) 


tan - tan x = sec a; — 1. 


2 sin 6 


cos 0 — sin 0 tan d 


= tan 2 6. 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. 

Ill 


1 . 


. MOULTON 

i 

\ 











- 



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NAME 


DATE 


SECTION 


87. Prove one of the following identities: 

(a) cos 2 0 — sin 0 = (1 — 2 sin 0)(1 + sin 0). 

/s cos (x — y) , cos (x + y) 

{€) -^^ ^ = cos X cos y. 


(6) sin 2 a:+cos 2 a:+l = 2 cos x (sin a:+cos x). 

/jx , 2(sin X 4- cos x) 

(a) sec rc + CSC rc = ^4 

sin 2 X 


88. Prove one of the following identities: 

, X sin 4: X ct 1 

(a) -= 2 cos® X — cos x. 

4 sin X 

(c) cos 4 0 = 8 cos^ 0 — 8 cos® 0 + 1. 


(h) 

(d) 


sin {x + y) sin {x — y) = 2 sin x cos y. 

COS29 - cos^ ^ cos9- 1. 

2 cos 0 + 1 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

113 










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NAME 


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89. Prove one of the following identities: 

(a) tan x sin x = -cos_^^ 

2 cos X 

(c) sin 3 a; = 3 sin a; — 4 sin® x. 


(&) 


1 — cos A 
1 + cos A 



1 . 


(d) cot a: + CSC a; 


2 cos X + cos 2 a; + 1 
sin 2 X 


90. Prove one of the following identities: 

(a) cos® X — cos 3 X = 3 sin^ x cos x. 

/ X 4 sin® A _ _ sec® A ^ 

sin® 2 A CSC® A 


(&) 

{d) 


i cos (x — y) — ^ cos (x y)= sin x sin y. 
/cot A\® _ /tan A\® _ sin 4 A 
Vcsc A/ Vsec A/ 2 sin 2 A 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

115 



















NAME 


DATE 


SECTION 


91. Prove one of the following identities: 

(a) COS {x + y) cos {x — y) = cos^ x — sin^ y. 

/, X . o 4 1 — cos 2 A 

(6) sm 2 A = ---• 

tan A 

(c) 8 cos^ A — 3 = COS 4 A + 4 cos 2 A. 

{d) tan 2 A (1 + tan A) = -^--• 

cot A — 1 


92. Prove one of the following identities: 

f \ sin (x — y) 

(a) cos xcosy =-^ • 

tan X — tan y 

(c) sin {x + y) sin {x — y) = sin^ x — sin^ y. 

(h) tan X + tan y = V). 

COS X COS y 

(d) COS 3 a: = 4 cos® a; — 3 cos x. 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

117 

















NAME 


DATE 


SECTION 


93. Prove one of the following identities: 


(a) 8 sin^ 0 + 4 cos 20 = cos 40 + 3. 

/ N cot^ X , tan^ X _ ^ 

1 + cot^ X 1 + tan^ X 


(6) sin (x + y) sin 

(d) 2 _ 

1 + cos 2 a 


{x — y) — cos^ y — cos^ x. 
(1 — cos 2 aY _ 
sin^ 2 a 


94. Prove one of the following identities: 

(a) sin a: + sin 3 a; = 2 sin 2 x cos x. 

(c) cos 3 a; = cos a: — 2 sin 2 x sin x. 


(h) sin 5 a; — sin 3 a: = 2 cos 4 a; sin x. 

(d) cos 8 a: + cos 4 a: = 2 cos 6 a: cos 2 x. 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

119 
















NAME 


DATE 


SECTION 


95. Give the radian measures in terms of r for the angles in one of the following sets: 


(a) 45°, 60°, 32° 15', 123° 13'. 
(c) 20°, 72°, 17° 40', 111° 28'. 


Ans. 


TT TT 129 _ 7393 
4’ 3’ ^ 10800 


Ans. (6), (c), or (d) 


(h) 30°, 90°, 54° 10', 159° 17'. 
(d) 40°, 22^°, 35° 5', 145° 44'. 


96. For one of the sets of Exercise 95, express the answers in decimal form, given that 1° = .0174533 
radians. Give answers to four decimal places: 

Ans. (a) .7854, 1.0472, .5629, 2.1505. 

Ans. (6), (c), or {d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 
121 

















NAME 


DATE 


SECTION 


97. Find the measures in degrees of the angles given in radians in one of the following sets. Do not 
use tables: 


TT TT 14 TT . p 

(a) g, 2 , 5 . 1 . 5 . 

(A ’’’ ^ ^ 22 TT 

^ ^ 4' 3 ' 15 ’ ' ■ 

(5) 3 , 2.25. 

(d) J, P X, 2.5. 

^ 9 2 ’ 20 


Ans. (a) 30°, 90°, 504°, 85° 56' (to the nearest minute). 

Ans. ( 6 ), (c), or (d) 

98. For one of the following sets, express each angle in radian measure, using tables and giving results 
to four decimal places: 


(a) 23° 14', 57° 23', 141° 25'. 

(c) 10° 46', 72° 34', 160° 15'. 

Ans. (a) .4055, 1.0016, 2.4681. 

Ans. ( 6 ), (c), or (d) 

( 6 ) 17° 26', 64° 17', 117° 20'. 

(d) 25° 54', 78° 26', 149° 40'. 


EXERCISES IN PLANE TRIGONOMETRY 

COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

123 

















NAME 


DATE 


SECTION 


99. For one of the following sets, in which radian measures are given, change to measures in degrees 
and minutes, using tables: 

(a) .1000, 1.2025, 2.1030. (6) .3000, 1.1180, 2.3225. 

(c) .7000, .9426, 1.9320. {d) .5000, 1.440, 3.0075. 

Ans. (a) 5° 44', 68° 54', 120° 30'. 

Ans. (b), (c), or (d) 


100. In each of the following problems the radius of a circle and an arc on that circle are given. For 
one of these problems find the angle at the center subtended by the arc. Express the result first in radians, 
then in degrees and minutes, assuming the measurements of radius and arc exact : 

(a) Radius = 12 ft., arc = 15 ft. (b) Radius = 4 in., arc = 6 in. 

(c) Radius = 1.5 ft., arc = 2.5 ft. (d) Radius = 225 yd., arc = 150 yd. 

Ans. (a) .8 radians, 45° 50'. 

Ans. (b), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

125 




1 




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DATE 


SECTION 


r 


NAME 


101. /» wM the fotlomng prvhlems an angle at the center of a circle and the arc it intercepts are 
gtvtn. f or one c^f the problems find the radius of the circle. Assume that given measurements are exact: 

(a) Aiiglo « '20\ arc » 7 in. (6) Angle = 30° 15', arc = 203 ft. 

(c) Anglo « 142°, arc = 20 yd. (d) Angle = 109° 23', arc = 200 ft. 



Ans. (o) 20.05 (to 4 places). 
Ans. (5), (c), or (d) 




EXERCISES ET PLAlfE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

127 











wrm. 


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NAME 


DATE 


SECTION 


In the following problems the symbol sin~^ a denotes the inverse sine of a, that is, an angle whose 
sine is a ; similarly for the other inverse functions. If a is a given number between — 1 and + 1, 
the inverse sine of a has infinitely many values, but only one of these lies between — 90° and + 90°; 
we call this the principal value and denote it by Sin“^ a (note that the first letter is capitalized). 
Principal values of the other inverse functions are similarly indicated. To complete their definition 
we add that Sin”^ a, Csc“^ a, and Tan^^ a lie between — 90° and + 90°, while Cos“^ a, Sec“^ a, and 
Cot~^ a lie between 0° and 180°. 


102. For one of the following sets (6), (c), (d) complete the writing of the indicated equations inform 
similar to that used in set (a); 


(a) Sin“^ 1 


= 90° = J rad., Tan-i Vs = 60° = J rad.. 


Cos-1 = 135 ° = ^ rad., Sec-i (-1) = 180° = x rad. 


(6) Cos-10 


1 


Tan-i ( - 1 = 

V vs; 

(c) Tan-11 

(d) Cot-10 
Tan-i (-1) 


Cot-1 (-1) = 

Csc-i 2 

Sec-i 1 = 

Cos-(-f) = 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

129 








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NAME 


DATE 


SECTION 


103. For one of the following sets (6), (c), (d) complete the writing of the indicated equations inform 
similar to that used in set (a); 


(a) sin- -L= 

V2 135° 


tan-V3 = 2®®Il ±>i-360° = 


TT 

+ n*360°= ^ 
3 TT 

T 


. ±2 riTT rad., cos~^ (~1) = 180° ± n • 360° =7r±2 wtt rad., 


TT 

3 

2 TT 

X 


120 ° 1 


3 


±2n7rrad., sec-K-2)= , „ +n-360°=, 

^ ^ 240° J “ ^ 

3 


±2mr rad. 


(h) cos~^ 1 


cot-1 (-V3) = 


csc-i 2 


sin -1 (- 1 ) = 


(c) tan-i 1 


sec-i V2 


sin-1 Q 



(d) sec-i (—V2) = 


cot-1 (^_-v/3) = 


sin“i 1 



EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J, MOULTON 

131 















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NAME 


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SECTION 


104. For one of the following sets (6), (c), (d) complete the writing of the indicated equations inform 
similar to that used in set (a); 

(a) Sin-1.2000 = 11° 32' = .2013 rad., Tan-i 3.125 = 72° 15'= 1.2610 rad., 

Cos-1 (_ ,5260) = 121° 44' = 2.1247 rad., Cot-i (- .4156) = 112° 34' = 1.9647 rad. 

(5) Csc-i 1.5709 

Tan-i (- 1.500) = 

(c) Cos-1.1054 

Sin-1 (_ ,9120) = 


, Cos-1,7850 

, Sec-i (- 2.3205) = 
, Tan-1.2325 
, Cot-1 (-.2358) = 


(d) Sec-12.050 


Cot-1,0535 


Sin-1 (_ ,7923) 


Csc-i (- 1.5027) = 


5. Without using tables complete the writing of the indicated 
% form similar to that used in set (a); 

I sin Sin-1 2 = 2 ^ gin cos-i f =± h 

tan sec-i ^ 


equations in one of the sets (6), (c). 


¥ 

tan tan-i 1 


ec Cos-1 (— - 1 ) 
os cos-1 (— 1 ) 


= + -^ V 7 , CSC Cot-1 ^ _ 2) = VH. 

= , cos Sin-1 4 = 

= , sin cot-1 (—3) = 

= , cot Tan-i 3 = 


(d) cot cot 1 (— ^) = 

sec Cot -1 (— 4 ) = 


, sin tan ^ = t 

, cos Csc-i (— I) = 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

133 




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106. By using tables complete the writing of the indicated equations in one of the sets (h), (c), (d) in 
form similar to that used in (a); 

(a) sin Cos-1.5000 = sin 60° 0' = .8660, tan sin-i .2350 = tan , = + .2420, 

166 24 


cos Cot-1 (- 1.254) = cos 141° 26' = - 

1 ^ 0 ° '^4' 

- .7819, sec tan-i (- .4266) = sec _230 g/ = + 1-087. 

( 6 ) cos Sin -1 ^5000 = cos = 

, cot sec-i 2.325 = cot = , 

sinTan-i (— 2.423) = sin = 

, CSC cos-1 (- _ .2796) = esc = 

(c) sin Sec-i 2.000 = sin = 

, tan cos-1.4562 = tan = , 

cos Csc-i (— 1.573) = cos = 

, cot sin -1 (_ .7g92) = cot = 

(d) cos Tan-i 1.000 = cos == 

, sec csc-i (3.027) = sec = ^ 

sinSec-i(— 1.213) = sin = 

, csctan-i(— .2077) = esc = 


107. Without using tables complete the writing of the indicated equations in one of the sets (5), (c), 
(d) in form similar to that used in set (a); 

(a) sin (90° + Sin-i f) = cos Sin-i f = |-, tan (180° - Sin-i =- tan Sin-i = - rV- 


(6) tan (90° - Cot-i (- 1)) = 

, cos (180° + Cot-1 f) = 

(c) cos (90° — Tan-i ( — f)) = “ 

, cot (180° +Sin-1 (-1)) = 

(d) sin ^90° + Cos-1 == 

, cos (180° — Tan-i = = 


EXERCISES IN PLANE TRIGONOMETRY 

COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

135 









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108. Find all solutions of one of the following equations, giving answers both in degrees and in radians: 


(a) V 2 sin 0 + 1 = 0. 

(c) 2 cos d +V 3 = 0. 

Ans. (a) “225l}±"'360° 
Ans. (6), (c), or {d) 


(h) 1 + tan 0 = 0. 
(d) Vs + cot 0 = 0. 


4 
5 TT 

T 


± 2nir rad. 


109. Find all solutions of one of the following equations, giving answers both in degrees and in radians: 

(a) 3 cot^ 0 — 1=0. (6) 4 sin^ 0 — 3 = 0. 

(c) COS0 = sin0. 

TT 2 T 

+ 2 riTT rad. 


(d) sin 0 + cos 0 = 0. 


. , ^ 60°, 120° 1 , OAAO 3’ 3 

^ ^ 240°, 300° J 4 TT 5 TT 


3 ’ 3 


Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

137 






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NAME 


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110. For one of the following equations find all solutions such that 0 ^ a; < 360° : 

(a) sin 2 X + cos x = 0. (6) cos 2 x — 1 = 2 sin x. 

(c) cos 2 X + 1 = 2 cos X. (d) sin 2 x + sin x = 0. 

Ans. (a) 90°, 210°, 270°, 330°. 

Ans. (b), (c), or (d) 


111. For one of the following equations find all solutions such that 0 ^ x < 180 : 


(a) cos X — cos 3 X = 0. 
(c) sin 3 X — sin x = 0. 

Ans. (a) 0°, 90°. 

Ans. (6), (c), or (d) 


(b) sin 2 X + sin 4 X = 0. 
(d) cos 2 X + cos 4 X = 0. 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

139 







. JTTACt 


9MAM 


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SECTION 


112. For one of the following equations find all solutions such that — 90° ^ x ^ 90° : 

(a) sin (x + 30°) = cos x. {h) sin (a; + 45°) = sin {x — 45°). 

(c) tan {x + 45°) = tan (45° — x). (d) sec (x + 60°) = sec x. 

Ans. (a) 30°. 

Ans. (6), (c), or {d) 


113. For one of the following equations find all solutions such that 0 ^ x ^ 360° : 

(a) CSC X + 2 sin x = 3. (5) 2 cos^ x + 3 sin x = 0. 

(c) 2 sin^ X — 3 sin x cos x — 2 cos^ x = 0. (d) cot^ x — 4 cot x = — 3. 

Ans. (a) 30°, 90°, 150°. 

Ans. (6), (c), or {d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

141 





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114. For one of the following equations find all solutions such that 0 ^ x ^ 360° : 

(a) 6 sec a: — 6 cos x = 5. (b) sec^ x = 3 — tan x. 

(c) sec a; + 2 cos x = 3. (d) csc^ a: = 2 cot a; + 4. 

Ans. (a) 48° 11', 311° 49'. 

Ans. (6), (c), or (d) 


115. For one of the following equations find all solutions such that — 90° ^ x ^ 90° : 
(a) 2 cos 2 rc + 7 sin a; = 0. (&) 3 cos 2 a: - 5 cos x = 3. 

(c) 1 + cos2 - - 2 cos2 X = 0. (d) sin2 ^ = e cos^ x. 

Ans. (a) - 14° 29'. 

Ans. (b), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

143 






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116. For one of the following equations find all solutions such that 0^0^ 180° : 

(a) sin 6 — sin 3 0 = cos d + cos 3 6. (b) cos 20 + cos 40 = sin 3 0. 

(c) sin 0 + sin 2 0 + sin 30 = 0. (d) sin 0 — sin 70 = cos 4 0. 

Ans. (a) 45°, 135°. 

Ans. (6), (c), or (d) 


117. For one of the following equations find all solutions such that 0^0^ 180° : 

(a) tan (0 + 45°) = 1 + tan 0. (&) tan (0 + 45°) = - tan (0 - 45°). 

(c) cos (0 + 60°) + cos (0 - 60°) = 0. {d) sin (0 + 30°) - sin (0 - 30°) = 0. 

Ans. (a) 0°, 135°, 180°. 

Ans. (5), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

145 





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118. For one of the following equations find all solutions such that — 90° ^ a; ^ 90° : 
(a) cos^ a; + sin a; + 1 =0. (b) cos a: + sin a: = 1. 

(c) sin I + cos X — 1 = 0. (d) tan ^ + sin ^ = 0. 

Ans. (a) — 90°. 

Ans. (6), (c), or (d) 


119. For one of the following equations find all solutions such that — 90° ^ x ^ 90° : 
(a) 2 sin 0 - cos 0 = 1. (fi) sin 0 + 3 cos 0 = 3. 

(c) sin 0 + cos 0 = i. (d) 5 sin 0 - 5 cos 0 = 1. 

Ans. (a) 53° 8'. 

Ans. (h), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

147 







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120. Given VlO = 10 ^ = 3.1623, </l0 = 10 = 1.7783, 

^10 = 10 = 1.3335, VlO = 10 0«25 = 1.1548, 


fov one of the sets (5), (c), or {d) cornjplete the equations as shown in {a): 

(a) log 31.623 = 1.5, log .11548 = - 1 + .0625, 

log 316.23 = 2.5, log .017783 = - 2 + .25. 

(31.623 = lO^ X 10-^ = 10^ ®, .-. log 31.623 = 1.5, etc.) 


(b) log 177.83 
log 11.548 

(c) log .013335 = 

log .13335 

(d) log 11,548 

log .00017783 = 


, log 13.335 
, log .031623 = 

, log .0017783 = 

, log 133,350 = 

, log .00011548 = 
, log 3,162,300 = 


« 


exercises in plane trigonometry 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

149 













NAME 


DATE 


SECTION 


121 . Given VlO = 10 ^ = 3.1623, ^10 = 10 = 1.7783, 

7T0 = 10 125 1.3335, Vio = 10 0525 = 1.1548, 

for one of the sets (6), (c), or (d) complete the equations as shown in (a) .* 


(a) log 31.623 

= 1.5000, 

(6) log 1333.5 = 

log .31623 

= - 1 + .5000, 

log 1.3335 = 

log 3162.3 

= 3.5000, 

log 133.35 = 

log 31623 

= 4.5000, 

log 1,333,500 = 

log 316,230 

= 5.5000. 

log 13.335 = 

(c) log .011548 

= j 

(rO log 17.783 = 

log 1154.8 

“ J 

log 1778.3 = 

log 115.48 

“ ) 

log 17,783 = 

log 115,480 

“ 7 . 

log .17783 = 

log 1,154,800 

= 

log 177,830 = 

122. Given log 2 = 0.30103, log 3 = 0.47712, hut using no other tables, find the logarithms required 
in one of the sets (5), (c), or (d) by using the fundamental laws of logarithms: 

(a) log 4 =2 log 2 = 0.60206, 

log 2 = log 2 - log 3 = 9.82391 - 10, 

log V2 = -I- log 2 = 0.15052, 

log 5 = log 10 — log 2 = .69897. 

(b) log 9 = 

? 

log 6 = } 

log ^3 = 

? 

log 15 = 

(c) log I = 

J 

log '^2 = > 

log 8 = 

y 

log 18 = 

(d) log 12 = 

y 

log f = J 

log 27 = 

7 

log ^3 = 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

151 


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123. Given VTo = lO-^ = 3.1623, 

^ = 10 125 = 1 . 3335 , 
find the values of the logarithms indicated 

(а) log (3.1623)3 = 1.5000, 

(б) log (13.335)2 = 

(c) log (115.48)3 = 

(d) log (1778.3)^ = 


A^IO = 1025 = 1.7783, 

VIO = 10 0625 = 1.1548, 

in one of the sets (6), (c), or (d): 

log ^31.623 = .5000. 

log VmM = 

log -^11.548 = 

log 'v/17,783 = 


124. Find the logarithms indicated in one 
in (6), 6-place logarithm tables in (c) and (d): 


(a) 

log 25.1 = 

1.3997, 


log 757 = 

2.8791, 

(&) 

log 568 = 



log .862 = 


(c) 

log 556.4 = 



log .6837 = 


id) 

log .1257 = 



log 9.275 = 



the sets (h), (c), or (d), using 4-place logarithm tables 

log .5564 = 9.7454 — 10, 

log 6.891 = 0.8383. 

log 7863 = , 

log .05648 = 

log .0056812 = , 

log 85.719 = 

log 100.25 = I 

log 41.623 = 


of 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

153 








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125. Using ^-pZace tables in (a) and (b), and 5-place tables in (c) and (d), find the numbers whose 
logarithms are: 

(а) 9.9222 - 10, 0.6325, 2.6000, 1.8725. 

(б) 1.8075, 8.4440 - 10, 0.7446, 2.2720. 

(c) 2.56170, 3.83607, 9.56282 - 10, 0.35128. 

{d) 9.25188 - 10, 0.68984, 5.44643, 2.84840. 

Ans. (a) .8360, 4.290, 398.1, 74.56. 

Ans. (6), (c), or {d) 


126. Making use of 4-place logarithm tables in (a) and (6) and of 5-place tables in (c) and (d) find 
the values of the following expressions. Carry results to four significant figures in (a) and (6), to five 
significant figures in (c) and (d): 

(a) 42.4 X 125, .4257 X 3.628. (6) 5.75 X .0817, 61.25 X .3751. 

(c) 4857 X 1.125, 2.6257 X .94525. (d) .06027 X 50.25, .81275 X 57.863. 

Ans. (a) 5300, 1.545. 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

155 







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127. Making use oj Jy-ylace logarithm tables find the values of the expressions in (a) or (6) to four 
significant figures, or using 6-place tables find the values of the expressions in (c) or (d) to five significant 
figures: 


(a) 

(c) 


3.45 
425 ’ 
8.3750 
.01250 ’ 


.06827 

-125 

^ ^ .0186’ 

5.565 

5153 ‘ 

6566 ■ 

23.625 

... 263.7 
® 6750 - 

560.31 

505.05' 

.0081752' 


Ans. (a) .008117, .00001325. 
Ans. (6), (c), or (d) 


128. Making use of 4-place logarithm tables find the values of the expressions in (a) or (b) to four 
significant figures, or using 5-place tables find the values of the expressions in (c) or (d) to five significant 
figures: 

(a) (6,125)’, ^2187. (5) (6.028)’, 

(c) (50.113)’, v'.012875. (d) ’7.69897, (3.1267)7 

Ans. (a) 229.8 (or 230.0), 1.479. 

Ans. (b), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

157 


















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129. In (a) or (6) find the value of the given expression to four significant figures hy use of 
logarithm tables, or in (c) or (d) find the value to five significant figures hy using 6-place tables: 


(a) J -5172 X .0251 
>'8.375 X .00234 
/685.2 X .008371 
^565.47 X 8.2345* 


(&) ij-- 

^7 


0056 X 8.107 


id) 


.0237 X 5071 
5687 X 75370 
.02563 X 1.5658' 


Ans. (a) ,8140 (by 4-place logs). 
Ans. (6), (c), or (d) 


130. Use 4-P^(^ce tables to find to four significant figures the value of the expression in (a) or (b), or 
use 5-place tables to find to five significant figures the value of the expression in (c) or {d): 


(«) -xi 

1.002541 

(b) ■ 

3/ .08623 

(.5612)3 

>(.005281)2 

(c) XI 

1 .43517 

(d) ' 

/.0058365 

(.26812)3 

>(.72573)3 

Ans. 

(a) .1199 (by 4-place tables). 




Ans. (6), (c), or {d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D, R. CURTISS AND E. J. MOULTON 

159 


























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SECTION 


131. Find the following logarithms, to four 
(a) log sin 25° 20' = 9.6313 - 10, 

log cos 16° 10' = 9.9825 - 10, 

log tan 43° 18' = 9.9742 - 10. 

(c) log cos 15° 8' = , 

log sin 35° 43' 

log tan 40° 25' 15" = 


places in (a) or (6), or to five places in (c) or (d): 
(h) log sin 5° 10' = 

log cot 27° 30' 
log cos 8° 12' 

(d) log cot 28° 15' = 

log tan 15° 47' = 

log cos 31° 18' 45" = 


9 


132. Find the following logarithms to four places in (a) or (6), or to five places in (c) or {d): 


(a) log sin 116° 20' 
log tan 87° 50' 
log sin 47° 12' 

(c) log cot 48° 15' 
log sin 167° 17' 
log cos 75° 18' 10" 


= 9.9524 - 10, 
= 1.4221, 

= 9.8655 - 10. 


(6) log sin 105°10' = 

log cot 65° 40' = 

log cos 71° 15' = 

{d) log sin 109° 29' = 

log tan 58° 51' 20" = 
log cos 69° 55' = 


133. Fmd the acute angle A (by use of tables in (a) or (b), by use of 5-place tables in (c) or 

{d)) in one of the following sets of equations: 

(a) log sin A = 9.4177 - 10, 
log cot A = 0.2928, 


log cos A = 9.9700 — 10. 
(c) log cos A = 9.96190 — 10, 
log tan A = 9.65636 — 10, 
log sin A = 9.65600 — 10. 
Ans. (a) 15° 10', 27° O', 21° 4'. 
Ans. (b), (c), or (d) 


(b) log cos A = 9.9368 — 10, 
log sin A = 9.6007 — 10, 
log tan A = 9.7075 — 10. 
(d) log tan A = 9.73416 - 10, 
log sin A = 9.59720 — 10, 
log cot A = 0.56348. 


134. Find the acute angle A (by use of Jrplace 
or (d)) in one of the following sets of equations: 

(a) log sin A = 9.9453 — 10, 
log tan A = 0.5331, 
log cos A = 9.8143 — 10. 

(c) log tan A = 0.06211, 

log sin A = 9.92449 — 10, 
log cot A = 9.70474 - 10. 


tables in (a) or (6), or by use of 6-place tables in 

(b) log cos A = 9.7380 — 10, 
log cot A = 9.7009 — 10, 
log sin A = 9.9427 — 10. 

(d) log cot A = 9.92894 - 10, 
log sin A = 9.95960 — 10, 
log tan A = 0.20710. 


(c) 


Ans. (a) 61° 50', 73° 40', 49° 18'. 
Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 
161 














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NAME 


DATE 


SECTION 


135. Solve and check one of the right triangles whose given parts are as follows. Use J^-place loga¬ 
rithms in (a) and (&), 5-place tables in (c) and {d). In each case C = 90°. Fill in the outline below, 
drawing an accurate figure in the space indicated: 

(a) A = 56° 40', a = 256. (6) A = 27° 15', a = .138. 

(c) A = 47° 15', a = 25.625. (d) A = 15° 17' 30", a = .5682. 

Ans. (a) B = 33° 20', b = 168.3, c = 306.4. 

Data 

C = 90° A = a = 


Fig. (1 cm. = 


Estimates from figure 
c = B 


Formulas 

B = 90° — A, b = a cot A, c = — 

sin A 


Computation 
log a = 

(+) log cot A = 
log b = 

log a = 

(—) log sin A = 
log c = 


Solution 

B = 
b = 


c = 


Check 

b"^ = — a^ = {c — a){c + a) 

c — a = 
c + a = 
log (c - a) = 

(+) log (c + a) = _ 

log (c^ — a^) = 

2 log b = 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

163 











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NAME 


DATE 


SECTION 


136. Solve and check one of the right triangles whose given parts are as follows. Use ^-pZace tables 

in (a) and (h), 5-place tables in (c) and (d). In each case C = 90°. Fill in the outline below: 

(a) B = 43° 10', c = 125. (6) B = 32° 25', c = 3.84. 

(c) B = 27° 18' 30", c = 234.1. Id) B = 19° 27', c = 1.5793. 

Ans. (a) A = 46° 50', a = 91.17, b = 85.50 (by 4-place logs). 

Data 

C = 90° B = c = 

Estimates from figure 
A = a = b = 

Formulas 

Fig. (1 cm. = ) A = 90° — B, a = c cos B, b = c sin B. 


Computation 
log c = 

(+) log cos B = 
log a = 
log c = 

(+) log sin B = 
log b = 


Solution Check 

A = a^ = c^ — ¥ = (c — b){c b) 

c — b = 

a = c -H 6 = 

log (c - 6) = 
log (c + 5) = 
b = log (c^ — ¥) = 

2 log a = 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

165 













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NAME 


DATE 


SECTION 


137. Solve and check one of the right triangles whose given parts are as follows. Draw a careful 
figure and from it estimate the unknown parts of the triangle; make a complete outline for the computa¬ 
tion, solution, and check, before performing the details of computation. Use 4-place logarithms in (a) 
and (b), 6-place logarithms in (c) and {d). In each case C = 90°; 

(a) b = .150, c = .275. (6) a = 15.25, c = 31.50. 

(c) b = 231.50, c = 357.25. [d) a = 156.7, c = 256.85. 

Ans. (a) A = 56° 56', B = 33° 4', a = .2304 (by 4-place tables). 


138. Solve and check one of the right triangles whose given parts are as follows, using ^-pZace tables 
in (a) and (b), and 5-place tables in (c) and (d). Before performing details of computation construct a 
careful figure, estimate unknown parts, outline the computation, solution, and check. In each case 
C = 90°: 

(a) a = 15.45, b = 31.56. (6) o = .5875, b = 1.027. 

(c) a = 287.30, b = 150.88. (d) a = 4.5876, b = 5.6859. 

Ans. (a) A = 26° 5', B = 63° 55', c = 35.14 (by 4-place tables). 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

167 





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NAME 


DATE 


SECTION 


139. Solve and check one of the triangles whose given parts are as follows. Use 4-place tables in (a) 
and (h), 5-place tables in (c) and (d). Construct the figure as indicated and fill in the printed outline: 

(a) A = 47° 40', B = 76° 10', c = 23.70. (6) A = 56° 30', B = 45° 20', c = 65.50. 

(c) A = 58° 27', B = 65° 18', c = 1.5280. (d) A = 78° 35', B = 18° 43', c = 14.580. 

Ans. (a) C = 56° 10', a = 21.09, b = 27.70. 

Data 

A = B = c = 


C = 


Estimates from figure 

a = 6 = 


Fig. (1 cm. = ) 


Formulas 

C = 180° - (A + B), 


c sin A 
sin C ’ 


b = 


c sin B 
sin C 


Computation 

Solution 

Check 

A = 


b — a _ 

B = 


b a 

A + 5 = 

C = 

b — a = 

logc = 


b a = 

(—) log sin C = 


B - A = 
5 + A = 

log-T^ = 
sin C 


- A) = 

(+) log sin A = 


+ A) = 

log a = 

a = 

log (6 - a) = 

log-T^ = 
sin C 

(+) log sin B = 


(-) log {b + a) = 
1 b — a 
b a 

log 6 = 

b = 

log tan i (5 — A) = 


tan i (B + A) 


(-) log tan 1 (5 + A) = 

. ta n ^ (B — A) _ ' 


log 


tan i (5 + A) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

169 



















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NAME 


DATE 


SECTION 


140. Solve and check one of the triangles whose given parts are as follows. Use 4-pl(^ce logarithm 
tables in (a) and (6), 5-place tables in (c) and {d). Construct the figure as indicated, draw up a complete 
outline as in Exercise 139, then fill in: 

(a) A = 48° 20', C = 25° 20', h = 156.5. (6) A = 105° 40', C = 40° 10', h = 18.51. 

(c) A = 75° 15', C = 26° 17', b = .56780. (d) A = 57° 40', C = 48° 56', 6 = 35.735. 

Ans. (a) B = 106° 20', a = 121.8, c = 69.78 (by 4-place tables). 


Data 



A = 

C = b 



Estimates from figure 


B = 

a = c 



Formulas 

Fig. (1 cm. = ) 

B = 

, a = , c 

Computation 

Solution 

Check 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D- R. CURTISS AND E. J. MOULTON 

171 






I ftn iyS.v\qi\^ aA^ Jr>w^\vsi*.(/J .(£^1^ It* ^?^Sj»4 l^fSlS^-^ t(4y ^W> if*') lit mS4x>J 

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NAME 


DATE 


SECTION 


141 . Solve and check one of the triangles whose given parts are as follows. Use 4-place tables in 
(a) and (b), 5-place tables in (c) and (d). Construct a figure and draw up an outline, then fill in the 
outline: 

(а) B = 28° 15', C = 48° 17', b = 2.670. 

(б) A = 47° 18', B = 52° 46', a = 3.207. 

(c) B = 82° 27' 40", C = 26° 30' 30", a = 25.637. 

(d) A = 32° 10' 15", B = 60° 5' 18", b = .60873. 

Ans. (a) A = 103° 26', a = 5.485, c = 4.210 (by 4-place tables). 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

173 











NAME 


DATE 


SECTION 


142. Solve and check one of the triangles whose given parts are as follows. Use /(.-place tables in (a) 
and ( 6 ), 5-place tables in (c) and (d). Construct the figure as indicated and fill in the printed outline: 

(a) A = 35°, a = 14, 6 = 24. ( 6 ) A = 53° 10', a = 29.5, b = 35.5. 

(c) A = 27° 2 ', a = 85, 6 = 185. (d) A = 29° 6 ', a = .32, b = .64. 

Ans. (a) Bi = 79° 30', Ci = 65° 30', Ci = 22.21; = 100° 30', C 2 = 44° 30', C 2 = 17.11. 


Data 


A = 


B^ = 

B, = 


a = 


Estimates from figure 
Cl = 

C 2 = 


b = 


Cl 

C2 


sin B = 


Formulas 

Cl = 180° - (A + 5i), Cl = 
a sin A 


Fig. (1 cm. = ) 

Computation 
log 6 = 

(+) log sin A = _ 

log 6 sin A = 

(-) log a =_ 

log sin B = 

A + Bi = 

A + ^2 = 
log a = 

( —) log sin A = __ 

1 a 

log^ — r = 
sin A 

(+) log sin Cl =_ 

log Cl = 

1 Ct 

log ^J = 
sin A 

(+) log sin C 2 =_ 

log C 2 = 


B 2 = 180° - Bu C 2 = 180° - (A + E 2 ), C 2 = 


a sin C 2 
sin A 


Solution 


Check 

b — Cl ^ tan ^ (Bi — Ci) 
6 + Cl tan 1 {Bi + Ci) 


Bi = 

B 2 = 

Cl = 
C 2 = 


b — Cl — 
5 + Cl = 

log (6 - Cl) = 
(-)log (6 +Cl) = 


log 


b — Cl _ 


Cl = 


C2 = 


i(52 - C2) = 


’ 5 + Cl 

i(5i-Ci) = 

i(Si + Cl) = 
log tan i {Bi — Cl) = 
log tan i (B 2 — C 2 ) = . 

loD- ¥ (-Cl ~ Cl) _ 

^ tan i (52 - C 2 ) 

6 — C 2 = 

6 + C 2 = 

log (6 - C 2 ) = 
(-)log (6 + C2) =. 
1 ■“ C2 

0 + C2 

, i (52 + C2) = 
log tan ^ {B 2 — C 2 ) = 

( —) log tan i (52 + C 2 ) = 


log 


tan (52 — C 2 ) _ 
tan ^ (52 + C 2 ) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 19>1 BY D. R. CURTISS AND E. J. MOULTON 

175 


























my^ 

3 v#o iji} 

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— y 







NAME 


DATE 


SECTION 


143. In one of the triangles whose given parts are as follows, find C only; do not check. Use 4-pl<^c^ 
tables in (a) and (h), 5-place tables in (c) and {d). Construct a figure, draw up an outline for the com¬ 
putation, then fill in the outline: 

(a) A = 120° 15', a = .5077, c = .3022. (5) B = 105° 15', a = 1.609, b = 5.407. 

(c) A = 130° 17' 30", a = 26.685, b = 10.428. {d) B = 150° 18' 25", a = 168.27, b = 325.62. 

Ans. (a) Cl = 30° 57', C 2 impossible. 

Ans. {b), (c), or {d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

177 





0 n^^Vti w\ .C^l 

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NAME 


DATE 


SECTION 


144. For one of the following sets of given parts show, hy carrying out the computation as far as 
necessary, that it is impossible to find another angle, so that no triangle exists with the given parts. Draw 
a figure: 

(6) B = 56° 18', a = 1.207, b = .8563. 

(d) A = 30° 25' 5", a - .10883, c = .32567. 


(a) A = 120° 17', a = 12.65, b = 36.85. 

(c) C = 65° 10' 20", a = 26.407, c = 18.635. 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

179 






v. r 




to "COX S» lao ^ «itSKt iWtiV V V'Vvf^X ‘iSi V w*o v>^ 

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NAME 


DATE 


SECTION 


145 . Solve and check one of the triangles whose given parts are as follows. Use 4-plo^^ tables in 
(a) and (6), 5-place tables in (c) and {d). Construct a figure and draw up an outline, then fill in the 
outline: 

(a) A = 47°, b = 56.5, a = 85.3. (6) A = 56° 10', b = 6.45, a = 12.75. 

(c) B = 27° 18', b = .3287, a = .2175. (d) A = 35° 25' 10", c = 87.324, a = 111.60. 

Ans. (a) B = 28° 58', C = 104° 2', c = 113.1 (by 4-place tables). 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

181 











NAME 


DATE 


SECTION 


146. Solve and check one of the triangles whose given parts are as follows. Use /(.-place tobies in 
(a) and (6), 5-place tables in (c) and {d). Construct the figure as indicated and fill in the printed outline: 

(a) a = 91.31, b = 75.68, C = 56° 12'. (b) a = .6271, b = .4585, C = 40° 17'. 

(c) a = 3.2170, b = 1.5267, C = 63° 18' 15". (d) a = 385.61, b = 256.33, C = 72° 2' 30". 

Ans. (a) A = 71° 50', B - 51° 58', c = 79.86 (by 4-place logs). 

Ans. (6), (c), or (d) 

Data 

a = b = C = 

Estimates from figure 

A = B = c = 


Fig. (1 cm. = ) 


Formulas 


A A B = 180° - C, 
tan ^ (A — B) = ^ tan ^ (A + B), 


a sin C 
sin A 


a = 
b = 
a — b = 
a A b = 
C = 
AAB = 
i(A A B) = 


Computation 

log (a - 6) = 
(-) log (a + 5) = 


(-1-) log tan ^{A A B) = 
log tan i (A — B) = 
i(A -B) = 
i(A A B) = 
log a = 
(+) log sin C = 
log a sin C = 
(—) log sin A = 
log c = 


Solution Check 

6 sin C =c sin B 
logb = 

(+) log sin C = _ 

log 6 sin C = 
logc = 

(-b) log sin B = _ 

log c sin 5 = 

A = 

B = 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

183 
















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DATE 


SECTION 


147. Solve and check one of the triangles whose given parts are as follows. Use Ji.-place tables in 
(a) and (6), 5-place tables in (c) and (d). Construct a figure, draw up an outline, then fill in: 

(a) A = 110° 12', b = 15.23, c = 25.75. (b) B = 125° 13', a = 1.534, c = 2.051. 

(c) C = 140° 30' 10", a = 20.563, b = 24.827. (d) A = 135° 18'20", b = .29703, c = .18560. 

Ans. (a) B = 24° 45', C = 45° 3', a = 34.13 (by 4-place tables). 

Ans. (6), (c), or {d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

185 






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NAME 


DATE 


SECTION 


148. Solve and check one of the triangles whose given 'parts are as follows. Use 4-vicice tables in 
(a) and (6), 5-place tables in (c) and {d). Construct the figure as indicated and fill in the printed outline: 

(a) a = 18.50, b = 16.00, c = 12.50. (6) a = 25.10, b = 18.20, c = 15.70. 

(c) a = 256, b = 350, c = 300. {d) a = 4.12, b = 3.90, c = 1.95. 

Ans. (a) A = 79° 56', B = 58° 22', C = 41° 42' (by 4-place tables). 

Ans. (6), (c), or (d) 

Estimates from figure 

A = B = C = 


Fig. (1 cm. = 


Formulas 


2s = a-\-b-{-c, 



r 2 = (g - a)(s - b)(s - c) ^ 




r 


s — c 


a = 
b = 
c = 
2s = 
s = 


Computation 
s — a = 
s-b = 

s — c = _ 

3 s — 2 s = 


log r = log r = 

(-) log (s - a) =_ (-) log (s - 6) = 

log tan ^ A = log tan ^ B = 

hB= 


log (s - a) = 
( + )log {s-b) = 
(-l-)log (s - c) = 
sum = 
(-) logs = 
log r^ = 2 log r = 

log r = 
(-)log (s - c) = 
log tan ^C = 

i C' = 


Solution and Check 
A = 

B = 

C = _ 

sum = 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

187 
































NAME 


DATE 


SECTION 


149 . Solve and check one of the triangles whose given parts are as follows. Use 4-plci'Ce tables in 
(a) and (6), 5-place tables in (c) and (d). Construct a figure, draw up an outline, then fill in: 

(a) a = .2253, b = .2074, c = .2735. (6) a = 2.837, b = 2.556, c = 2.103. 

(c) a = .10433, b = .05422, c = .12586. (d) a = 3.1275, b = 4.0768, c = 2.6755. 

Ans. (a) A = 53° 46', B = 47° 56', C = 78° 18' (by 4-place tables). 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

189 





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NAME 


DATE 


SECTION 


150. Solve and check one of the triangles whose given parts are as follows. Use tables in 

(a) and (6), 5-place tables in (c) and (d). Construct a figure, draw up an outline, then fill in: 

(a) a = 8.057, b = 12.350, c = 7.501. (b) a = 5.001, b = 7.755, c = 3.752. 

(c) a = 1.08261, b = .72568, c = .45362. (d) a = 200.52, b = 291.36, c = 200.52. 

Ans. (a) A = 39° 4', B = 105° 2', C = 35° 54' (by 4-place tables). 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

191 






P^iAoX 




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NAME 


DATE 


SECTION 


161. Using tables in (a) and (6), 5-place tables in (c) and (d), find the area of one of the 

triangles which have the following given parts: 

{Formula: Area = -I- 6c sin A = ^ab sin C = ^ ac sin B.) 

(a) A = 33° 10', 6 = 34.1, c = 28.5. (6) B = 105° 10', a = 267.5, c = 300.7. 

(c) C = 45° 27', a = 424.17, 6 = 536.85. {d) A = 32° 18', 6 = .73568, c = .95621. 

Ans. (a) 265.8 (by 4-place tables). 

Ans. (6), (c), or {d) 


152. Find the area of one of the triangles which have the following given parts, using 4-pl(^ce tables 
in (a) and (6), 5-place tables in (c) and {d): 

{Formula: Area =Vs(s — a){s — b){s — c).) 

(a) a = 32.56, 6 = 42.87, c = 51.65. (6) a = 3.685, 6 = 2.286, c = 2.433. 

(c) a = 40.213, 6 = 50.652, c = 52.531. {d) a = 120.63, 6 = 171.12, c = 95.33. 

Ans. (a) 695.7 (by 4-place tables). 

Ans. (6), (c), or {d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

193 





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NAME 


DATE 


SECTION 


153. Find the area of one of the triangles which have the following given parts, using 4-place tables 
in (a) and (b), 5-place tables in (c) and (d): 

(a) A = 32° 10', a = 18.31, b = 15.75. (6) A = 120° 20', a = 5.828, b = 3.612. 

(c) B = 104° 50' 45", b = 351.15, c = 200.75. (d) C = 65° 30' 30", b = 4812.3, c = 7826.2. 

Ans. (a) 124.2 (by 4-place tables). 

Ans. (6), (c), or {d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

195 
















NAME 


DATE 


SECTION 


154. Solve one of the following problems, using 4-place tables in (a) and (6), 6-place tables in (c) and 
(d): 

{a) A flagpole stands upon a level surface. From a point 150 ft. away an observer whose instru¬ 
ment stands 5 ft. above the surface of the ground finds the angle of elevation of the top to be 28° 10'. 
How high is the flagpole ? 

(6) An observer on the edge of a cliff finds the angle of depression of a boat to be 32° 10'. If 
the observer’s instrument is 250 ft. above the water level, how far is the boat from a point at water 
level directly beneath the observer? 

(c) Two boys on one bank of a river sight on each other and on a tree on the opposite bank. 
The angle between these lines of sight at one position is 90°, at the other is 37°, the distance between 
the boys is measured and found to be 125.2 ft. How wide is the river at this point? 

(d) If the radius of the earth (considered as spherical) is 3959 miles, what is the shortest distance 
through the earth between two points on the equator whose difference of longitude is 20° 10' 20"? 

Ans. (a) 85.32 ft. (by 4-place tables). 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

197 






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NAME 


DATE 


SECTION 


155. Find the answer to one of the following problems by solving one or more right triangles. Use 
4-place tables in (a) and (b), 5-place tables in (c) and (d): 

(а) On one side of a ditch the angle of elevation of the top of a wall on the other side is 45°. At 
a point 10 ft. straight back from this point the angle of elevation of the top of the wall is 40°. 
Find the height of the wall. 

(б) Two members of a surveying class sight on the same tower. If one measures the angle of 
elevation of the top as 41° 10' and his distance from the base as 123.2 ft., and the other finds an 
angle of elevation of 42° 30' and measures his distance as 119.5 ft., what is the difference in their 
measurements of the height of the tower? 

(c) Two observers 3 miles apart on a level plain sight on a balloon at an instant when it is directly 
above the line joining them. If the two angles of elevation are 20° 17' and 32° 15', how high is the 
balloon? 

(d) A pyramid is 150.50 ft. square at the base and 112.80 ft. high. Find the length of a slanting 
edge, and the angle it makes with the plane of the horizontal base. 

Ans. (a) 52.17 ft. (by 4-place tables). 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

199 








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NAME 


DATE 


SECTION 


156. Solve one of the following 'problems. Use 4-place tables in (a) and (h), 5-place tables in (c) 
and {d): 

(а) At one end of a lake the angle between the lines of sight to the other end of the lake and to a 
tree 150 yd. away is 47° 30'. At the other end of the lake the corresponding angle (sighting down 
the lake and at the same tree) is 23°. How far apart are the points of observation? 

(б) A pole 10 ft. tall is held upright on a slope. At a point 17 ft. down the slope the pole subtends 
an angle of 18°. What angle does the pole make with the ground ? 

(c) Two lighthouses are known to be 6000 ft. apart, one due north of the other. At a certain 
time an observer at each sights on a ship at sea. At one lighthouse the ship’s bearing is 
S 45° 30' E, at the other N 64° 20' E. How far is the ship from each lighthouse? 

{d) A tightly stretched guy-wire 25.234 ft. long makes an angle of 122° 10' 30" with the pole to 
which it is attached. A second wire is attached to the pole just 10 ft. higher up and anchored at 
the same point as the first. What is the angle between the two wires? 

Ans. (a) 361.8 yd. (by 4-place tables). 

Ans. (6), (c), or {d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 
201 







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NAME 


DATE 


SECTION 


157. Solve one of the following problems. Use 4-plci'Ce tables in (a) and (6), 6-place tables in (c) 
and {d): 

(а) A parallelogram has its non-parallel sides of length 5.128 in. and 4.105 in. respectively. If 
the angle between these sides is 65° 18', what is the length of the shorter diagonal of the parallelo¬ 
gram ? 

(б) What is the length of the longer diagonal of the parallelogram described in (a) ? 

(c) Two streets meet at an angle of 75° 15'. If a man owns a triangular corner lot having front¬ 
ages of 65.214 ft. and 85.926 ft. on the two streets, how much fencing will be needed to inclose it 
completely? 

(d) A pole standing on sloping ground is stayed by a tight wire rope 25.620 ft. long. If this rope 
makes an angle of 65° 30' 10" with the pole and of 38° 0' 30" with the ground, how high on the 
pole is the rope attached? 

Ans. (a) 5.050 in. (by 4-place tables). 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

203 









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NAME 


DATE 


SECTION 


158. Solve one of the following problems. Use 4-pl(^e tables in (a) and (b), 5-place tables in (c) 
and (d) : 

(a) How far apart are the tips of the hands of a clock at 8 o’clock if the hands are 10.62 in. and 
8.12 in. long respectively? 

(b) Find the intensity and direction of the resultant of two forces of 532.1 lb. and 465.5 lb. acting 
at an angle of 72° 10' to each other. 

(c) A lighthouse bears S 17° 15' W as seen from a ship. After the ship sails 7.5 miles on a course 
S 25° 10' E the lighthouse bears S 61° 18' W. How far was the ship from the lighthouse at each 
point of observation? 

(d) The three sides of a triangle are 32.412 in., 22.650 in., 16.427 in., respectively. Find the 
angle at which the median to the longest side meets that side, and the angles it makes with the other 
two sides. 

Ans. (a) 16.28 in. (by 4-place tables). 

Ans. (b), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

205 









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NAME 


DATE 


SECTION 


159. Solve one of the following problems: 

(а) A man standing on a building 70 ft, high (including the height of the instrument) observes 
two points A and B on the level surface on which the building stands. The angle of depression of 
A is 27° 15', that of B is 42° 18', The angle subtended by the distance AB is 67° 18'. Find the 
length of AB. 

(б) Ship A leaves a certain port on a course S 42° E and makes an average speed of 10.35 miles 
per hour. Ship B leaves the same port exactly 2 hours later and sails on a course S 20° E at an 
average speed of 15.61 miles per hour. How long after A left port will B be directly south of A ? 
How far will each have sailed? 

(c) A street and B street meet at an angle of 65° 17'. The corner lot has a frontage of 65 ft. on 
A street and 85 ft. on B street. The next lot is trapezoidal, running from one street to the other, 
with a frontage of 75 ft. on A street. What is its frontage on B street, and what are the lengths of 
the two side lot lines? 

(d) A farmer fences off a triangular piece at the corner of a pasture. If he runs his fence from 
a point 28 yd. from the corner on one fence, uses 32 yd. of fencing, and ends at a point 25 yd. 
from the corner on the other fence, what was the angle of the original corner? At what angles will 
the new fence meet the old? What is the area enclosed? 

Ans. (a) 148.1 ft. (by 4-place tables). 

Ans. (b) (by 4-place tables) 

(c) or (d) (by 5-place tables) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

207 






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NAME 


DATE 


SECTION 


160. Solve one of the following problems: 

(а) A flagpole stands on the edge of a building. At a window directly opposite the angle of 
elevation of the top is 14° 10', the angle of depression of the bottom is 11°. At another window 
15 ft. lower and directly below the first the angle of elevation of the top is 32° 6'. Find the height 
of the flagpole and the width of the street. 

(б) Observers at two points A and B 5132 ft. apart sight on two positions S and S' of a ship. 
At 2 p.M. AABS = 38° 17', Z BAS = 112° 17'; at 2:15 p.m. Z ABS' = 118° 12', Z BAS' = 30° 14'. 
What was the ship’s average speed in miles per hour ? If .B is exactly SE from A what was the ship’s 
course between observations? 

(c) Two circles of radius 43.212 and 25.567 intersect, the distance between their centers being 
52.067. Find the angle between the tangents at the point of intersection, the length of the common 
chord, the angle subtended by the common chord as viewed from the center of each circle. 

(d) A square and a regular hexagon are inscribed in a circle of radius 10.000, with two vertices 
common. Find the total area of the triangular pieces cut off the hexagon by the square and of 
those cut off the square by the hexagon. 

Ans. (a) 17.88 ft., 40.01 ft. (by 4-place tables). 

Ans. (b) (by 4-place tables) 

(c) or (d) (by 5-place tables) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R, CURTISS AND E, J. MOULTON 

209 






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DATE 


SECTION 


161. For one of the sets (6), (c), or 

(d) complete the equations in the form shown in 

(a); 


(a) 

^ radians = 90°, 


^ radians = 135°, 


^ radians = 

O 

420°, 



45° = 1 radians, 


120° = ^ radians, 
o 


36° = ^ radians. 

5 


(h) 

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J 

^ radians = 

6 


^ radians = 
4 


9 


18° = 

radians. 

150° = 

radians. 

o 

o 

II 


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(c) 

^ radians = 

O 


TT radians = 


^ radians = 
4 




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radians. 

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radians. 

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radians. 

id) 

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4 


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5 

} 

radians = 

6 


J 


750° = 

radians. 

20° = 

radians. 

135° = 


radians. 


162. Prove one of the following identities: 

(a) cos^ a: - sin'^ re = cos 2 rr. (&) sin 2 rc tan re + cos 2 re = 1. 

(c) cot 0 - tan 0 = 2 cot 2 6. (d) sin (x + y) sin (x - y) = cos* y - cos* x. 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 
211 








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NAME 


DATE 


SECTION 


163 . For one of the following equations find all solutions such that 0 ^ a: < 360 : 


(a) 2 cos^ X + 3 sin a: = 3. 
(c) sin 2 a: — cos a; = 0. 

Ans. (a) 30°, 90°, 150°. 
Ans. (h), (c), or (d) 


(h) 2 sec^ X — tan a: = 2. 
(d) cos 2 X = cos a: — 1. 


164 . For one of the following equations find all solutions such that 0^0^ 180° : 

(a) tan 6 - tan (0 + 45°) +2 = 0. (&) 2 sin (0 + 60°) = cos 0 tan 60°. 

(c) sin 2 0 - tan 0 = cos 90°. {d) tan^ 0 - tan 0 = 6. 

Ans. (a) 22° 30', 112° 30'. 

Ans. (6), (c), or {d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. 

213 


f. MOULTON 





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NAME 


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SECTION 


165. Solve one of the following 'problems. The angle x is acute: 

(а) Given sin x = ■^, find tan x, cos ^ x, sin 2 x. 

(б) Given cos x = f, find sin x, tan 2 x, tan i x. 

(c) Given tan x = 2, find sin x, cos ^ x, tan 2 x. 

(d) Given sin x = f, find cos x, cos 2 x, tan ^ x. 

Ans. (a) .3535, .9856, .6285. 

Ans. (5), (c), or (d) 


166. Without referring to tables solve one of the following equations. Principal values are indicated 
by capital letters as in Exercise 102: 


(a) Sin-11 _j_ cos-1 ^ = |- 

(c) tan-i y = Csc-i 2 + Sin-i \ 

Ans. (a) iVs. 

Ans. (6), (c), or (d) 


(6) Tan-i 1 + Cot-i 1 + cos-i y = tt. 

(d) Cot-1 (- 1) - Sin-1 + sin-1 ^ = q. 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

215 






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DATE 


SECTION 


167. Solve one of the following problems, using right triangles: 

(а) If a wagon wheel with 8 spokes measures 21 inches from the center of the hub to the rim, how 
far apart are the ends of consecutive spokes ? 

(б) A pair of scissors measures 14.2 in. from screw to tip. What angle do the blades make when 
the tips are 10 in. apart? Assume that the angle of blades is that of lines from screw to tips. 

(c) What is the perimeter of a regular decagon inscribed in a circle of radius 10.62 in. ? 

(d) Find the angle of slope of an A-roof of a building 25 ft. 9 in. wide at the eaves, the ridgepole 
being 8 ft. 6 in. above the eaves. How long must rafters be to run from the ridgepole 6 in. past 
the eaves? 

Ans. (a) 16.07 in. (by 4-place tables). 

Ans. (6), (c), or (d) 


168. Solve one of the following problems: 

(а) A tree makes an angle of 122° 10' with the hillside on which it stands. At a point 60 ft. 9 in. 
down the slope it subtends an angle of 26° 15'. How tall is the tree ? 

(б) A certain rock bed is known to dip at an angle of 34° 12' to the horizontal. The surface of 
the ground slopes in the same direction at an angle of 14° 10'. At a point 250 ft. down the slope 
from the outcrop how far must one drill vertically to reach this bed ? 

(c) From the top of a monument standing on a hill the angle of depression of an object on the 
plain below is 46° 15'. From the bottom of the monument the angle of depression is 37° 5'. If 
the monument is 95 ft. high, how far is the object observed from the top? From the bottom? 

(d) A headland bears S 22° W as seen from a ship. The ship sails 15 miles on a course S 31° E, 
and the headland then bears N 68° W. How far was the ship from the headland at each position? 

Ans. (o) 51.31 ft. (by 4-place tables). 

Ans. (6), (c), or (d) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

217 









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NAME 


DATE 


SECTION 


169. Solve one of the following 'problems: 

(а) A flagpole 51 ft. 9 in. high stands on the top of a mound, making an angle of 105° 10' with 
the slope at the base of the pole. What is the elevation of the sun when the shadow of the flagpole 
is 58 ft. 3 in. long? Assume that the mound is conical. 

(б) From an elevated position the distances to the bases of two trees A and B are observed v;ith 
a range flnder to be 4256 yd. and 3171 yd., respectively. Measurement has shown these trees to 
be 3826 yd. apart. What angle should the distance AB subtend at the point of observation? 

(c) Two circles of radius 10.562 ft. and 18.213 ft., respectively, intersect, their common chord 
being 15.115 ft. in length. Find the distance between centers and the angle between the tangents 
at the points of intersection (two solutions). 

(d) Two forces, of 275.18 lb. and 361.02 lb., respectively, are balanced by a single force of 221.71 
lb. At what angle are the two forces acting? What angle does the equilibrating force make with 
each? 

Ans. (a) 41° 18' (by 4-place tables). 

Ans. (6) (by 4-place tables) 

(c), (d) (by 5-place tables) 


EXERCISES IN PLANE TRIGONOMETRY 


COPYRIGHT 1931 BY D. R. CURTISS AND E. J. MOULTON 

219 




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